{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "4cdbbb7f-08ac-4149-99d8-0c204024c70e",
   "metadata": {},
   "source": [
    "## 平面几何"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4e700c97-d2a6-4de8-a64b-fe31cc086a14",
   "metadata": {},
   "source": [
    "以下是椭圆、双曲线和抛物线相关信息的表格对比：\n",
    "|类型|椭圆|双曲线|抛物线|\n",
    "|--|--|--|--|\n",
    "|定义|平面内与两个定点$F_1,F_2$的距离之和等于常数2a（大于$|F_1F_2|=2c$）的点的轨迹|平面内与两个定点$F_1,F_2$的距离之差的绝对值等于常数2a（小于$|F_1F_2|=2c$且大于零）的点的轨迹|平面内，到定点$F$与定直线$l$的距离相等的点的轨迹|\n",
    "|标准方程|焦点在$x$轴：$\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}} = 1(a>b>0)$<br>焦点在$y$轴：$\\frac{y^{2}}{a^{2}}+\\frac{x^{2}}{b^{2}}=1(a > b > 0)$|焦点在$x$轴：$\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1(a>0,b>0)$<br>焦点在$y$轴：$\\frac{y^{2}}{a^{2}}-\\frac{x^{2}}{b^{2}} = 1(a>0,b>0)$|焦点在$x$轴正半轴：$ y^{2}=+2px(p>0)$ <br>焦点在$x$轴负半轴：$y^{2}=-2px(p>0)$ <br>焦点在$y$轴正半轴：$x^{2}=+2py(p>0)$ <br>焦点在$y$轴负半轴：$x^{2}=-2py(p>0)$|\n",
    "|焦点|焦点在$x$轴：$F_1(-c,0),F_2(c,0)$<br>焦点在$y$轴：$F_1(0,-c),F_2(0,c)$，其中$c^{2}=a^{2}-b^{2}$|焦点在$x$轴：$F_1(-c,0),F_2(c,0)$<br>焦点在$y$轴：$F_1(0,-c),F_2(0,c)$，其中$c^{2}=a^{2}+b^{2}$|焦点在$x$轴正半轴：$(\\frac{p}{2},0)$<br>焦点在$x$轴负半轴：$(-\\frac{p}{2},0)$<br>焦点在$y$轴正半轴：$(0,\\frac{p}{2})$<br>焦点在$y$轴负半轴：$(0,-\\frac{p}{2})$|\n",
    "|顶点|$(\\pm a,0),(0,\\pm b)$或$(0,\\pm a),(\\pm b,0)$|$(\\pm a,0)$或$(0,\\pm a)$|$(0,0)$|\n",
    "|系数关系|$c^{2}=a^{2}-b^{2}$|$c^{2}=a^{2}+b^{2}$|无类似$a,b,c$的系数关系，$p$决定焦点位置和开口大小|\n",
    "|离心率|$e=\\frac{c}{a}$，$0<e<1$|$e=\\frac{c}{a}$，$e>1$|$e = 1$|\n",
    "|准线|焦点在$x$轴：$x=\\pm\\frac{a^{2}}{c}$<br>焦点在$y$轴：$y=\\pm\\frac{a^{2}}{c}$|焦点在$x$轴：$x=\\pm\\frac{a^{2}}{c}$<br>焦点在$y$轴：$y=\\pm\\frac{a^{2}}{c}$|焦点在$x$轴正半轴：$x = -\\frac{p}{2}$<br>焦点在$x$轴负半轴：$x=\\frac{p}{2}$<br>焦点在$y$轴正半轴：$y = -\\frac{p}{2}$<br>焦点在$y$轴负半轴：$y=\\frac{p}{2}$|\n",
    "|渐近线|无|焦点在$x$轴：$y=\\pm\\frac{b}{a}x$<br>焦点在$y$轴：$y=\\pm\\frac{a}{b}x$|无|"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7406ae32-f1bc-49a4-bf9c-5ac75a7baa98",
   "metadata": {},
   "source": [
    "平面内，到定点$F$的距离与到定直线$l(F\\notin l)$的距离之比等于定值$e$的点的集合。$0\\lt e\\lt 1$时，轨迹是椭圆；$e\\gt 1$时，轨迹是双曲线;$e=1$时，轨迹是抛物线。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d35cce2f-b455-4964-8eeb-f31cb0e0fcb8",
   "metadata": {},
   "source": [
    "平面直线方程的五种形式如下：\n",
    "\n",
    "|名称|条件|方程形式|不能表示的直线|\n",
    "|--|--|--|--|\n",
    "|点斜式|已知直线上一点$(x_0,y_0)$和直线的斜率$k$| $y - y_0=k(x - x_0)$|垂直于$x$轴的直线（斜率不存在的直线）|\n",
    "|斜截式|已知直线的斜率$k$和在$y$轴上的截距$b$| $y = kx + b$|垂直于$x$轴的直线（斜率不存在的直线）|\n",
    "|两点式|已知直线上两点$(x_1,y_1)$，$(x_2,y_2)$，且$x_1\\neq x_2$，$y_1\\neq y_2$| $\\frac{y - y_1}{y_2 - y_1}=\\frac{x - x_1}{x_2 - x_1}$|垂直于坐标轴的直线|\n",
    "|截距式|已知直线在$x$轴和$y$轴上的截距分别为$a$，$b$，且$a\\neq0$，$b\\neq0$| $\\frac{x}{a}+\\frac{y}{b}=1$|过原点的直线以及垂直于坐标轴的直线|\n",
    "|一般式|无特定限制条件| $Ax + By + C = 0$（$A$，$B$不同时为$0$）|无|\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2af8683d-b292-4710-a1e5-693f8ac6d04f",
   "metadata": {},
   "source": [
    "## 高数"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f33b628-38ab-41eb-9baa-8f31fad9e4bf",
   "metadata": {},
   "source": [
    "### 1.极限"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "75bc2987-e6fa-40ff-b323-7c5860b4ae36",
   "metadata": {},
   "source": [
    "#### (1)求极限\n",
    "\n",
    "$$\\lim\\limits_{n\\rightarrow \\infty} \\sqrt{n}(\\sqrt{n+1} - \\sqrt{n})$$\n",
    "\n",
    "原式可化为$$\\lim\\limits_{n\\rightarrow \\infty} \\frac{\\sqrt{n}}{(\\sqrt{n+1} + \\sqrt{n})}$$\n",
    "上下同除以$\\sqrt{n}$最终结果为$\\frac{1}{2}$\n",
    "\n",
    "参考P57"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9d995b0d-5d84-48ed-b692-aa76baeb2708",
   "metadata": {},
   "source": [
    "#### (2)设$a_i>0,i=1,2,\\cdots,m$，求$\\lim\\limits_{n\\rightarrow\\infty}(a_1^n+a_2^n+\\cdots+a_m^n)^{\\frac{1}{n}}$\n",
    "**(夹逼定理)** 设$\\lim\\limits_{n\\rightarrow \\infty}x_n = a$,$\\lim\\limits_{n\\rightarrow \\infty}y_n=a$当$\\exists N\\in N^*$当$n\\gt N$时，有$x_n\\le z_n\\le y_n$,则$\\lim\\limits_{n\\rightarrow \\infty}z_n = a$\n",
    "\n",
    "参考P57"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd4e4f55-d83f-4d6e-9136-972264920a17",
   "metadata": {},
   "source": [
    "**重要结论**\n",
    "\n",
    "$\\lim\\limits_{n\\rightarrow \\infty} \\sqrt[n]{m} =1(m\\gt 0),\\lim\\limits_{n\\rightarrow \\infty} \\sqrt[n]{n} =1$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b6663031-3d8d-4fc1-b81e-b9427a4b0877",
   "metadata": {},
   "source": [
    "**柯西准则**\n",
    "\n",
    "设函数$f(x)$在点$x_0$的某一去心邻域$U_0(x_0,\\sigma_0)(\\sigma_0\\gt 0)$内有定义，则$\\lim\\limits_{x\\rightarrow x_0}f(x)$存在的充要条件是$\\forall \\epsilon\\gt 0 ,\\exists\\sigma\\gt0$,当$x^{'},x^{''}\\in U_0(x_0,\\sigma)$时，有$\\vert f(x^{'})- f(x^{''})\\vert\\lt\\epsilon$。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "be8f0697-f119-4aed-b7d2-aaf8ef117fb1",
   "metadata": {},
   "source": [
    "#### (3)设$x_1=\\sqrt{2},x_{n+1} = \\sqrt{2+x_n}$,证明数列$\\{x_n\\}$收敛，并计算极值。\n",
    "**（单调有界定理）：** 若数列$\\{x_n\\}$是单调数列并且有界，则数列$\\{x_n\\}$一定收敛。\n",
    "\n",
    "参考P58"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "237992fb-9b93-4e1c-8826-babec0a7f3c3",
   "metadata": {},
   "source": [
    "(4)求$\\lim\\limits_{x\\rightarrow 0}\\frac{e^{\\tan x}-e^{sin x}}{\\tan x - \\sin x}$\n",
    "\n",
    "等价无穷小因子替换求极限(依据极限的乘法运算法则，而极限的加法运算法则不能使用)：\n",
    "$$\\lim\\limits_{x\\rightarrow 0}\\frac{e^{\\tan x}-e^{sin x}}{\\tan x - \\sin x} = \\lim\\limits_{x\\rightarrow 0}\\frac{e^{sin x}(e^{\\tan x-\\sin x}-1)}{\\tan x - \\sin x}$$\n",
    "再由于$e^{\\tan x-\\sin x}-1\\sim \\tan x-\\sin x$可得最终结果为1。\n",
    "\n",
    "P65"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e96d7ab6-01a8-433e-b8ae-062a3002a64d",
   "metadata": {},
   "source": [
    "(5)求极限$\\lim\\limits_{x\\rightarrow 0}\\frac{x^{\\frac{3}{2}}\\sin \\frac{1}{x}}{\\sin x}$的值\n",
    "\n",
    "利用$\\sin x \\sim x$可以将原式化为$\\lim\\limits_{x\\rightarrow 0}x^{\\frac{1}{2}}\\sin\\frac{1}{x}$最终结果为0。\n",
    "\n",
    "P65"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ffb89989-954b-4c62-bae4-140a32677b0f",
   "metadata": {},
   "source": [
    "（6）讨论曲线$y = \\frac{x^3}{x^2-1}$的渐近线。\n",
    "\n",
    "渐近线的分类及求法：\n",
    "\n",
    "- 水平渐近线。若$\\lim\\limits_{x\\rightarrow -\\infty}f(x) = a$或$\\lim\\limits_{x\\rightarrow +\\infty}f(x) = a$，则称$y=a$是$y=f(x)$的一条水平渐近线。\n",
    "- 垂直渐近线。若$\\lim\\limits_{x\\rightarrow c^+}f(x)$和$\\lim\\limits_{x\\rightarrow c^-}f(x)$至少有一个是无穷大，则称$x=c$为$y=f(x)$的垂直渐近线。\n",
    "- 斜渐进线。若$\\lim\\limits_{x\\rightarrow +\\infty}\\frac{f(x)}{x}=k$存在且不为零(为零时为水平渐近线)，$\\lim\\limits_{x\\rightarrow +\\infty}[f(x)-kx]=b$也存在(或者$\\lim\\limits_{x\\rightarrow -\\infty}\\frac{f(x)}{x}=k$存在且不为零，$\\lim\\limits_{x\\rightarrow -\\infty}[f(x)-kx]=b$也存在)则称$y=kx+b$是$y=f(x)$的斜渐近线。\n",
    "\n",
    "所以上述曲线有：\n",
    "- 垂直渐近线：$x=1$和$x=-1$\n",
    "- 斜渐近线：y = x\n",
    "\n",
    "P66"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f417ba6-1d64-45af-a45e-dae265a0d6f7",
   "metadata": {},
   "source": [
    "以下是一些常用的**等价无穷小量**，当$x\\to0$时：\n",
    "1. $x\\sim\\sin x\\sim\\tan x\\sim\\arcsin x\\sim\\arctan x$\n",
    "2. $1 - \\cos x\\sim\\frac{1}{2}x^{2}$\n",
    "3. $e^{x}-1\\sim x$\n",
    "4. $a^{x}-1\\sim x\\ln a$（$a>0,a\\neq1$）\n",
    "5. $\\ln(1 + x)\\sim x$\n",
    "6. $(1 + x)^{\\alpha}-1\\sim\\alpha x$（$\\alpha\\neq0$）\n",
    "7. $\\sqrt{1 + x}-1\\sim\\frac{1}{2}x$\n",
    "\n",
    "等价无穷小量在求极限的运算中经常被使用，可以通过等价替换来简化计算，但需要注意替换的条件是在乘除运算中一般可以直接使用，在加减运算中使用时需要谨慎，避免出现错误。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "28275987-38a1-4fdf-a762-3797da95ac28",
   "metadata": {},
   "source": [
    "### 2.函数连续性"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0dc101be-bf90-4ba9-8afe-5507f3eed974",
   "metadata": {},
   "source": [
    "**【函数连续】：** 若对$\\forall \\epsilon \\gt 0,\\exists \\sigma\\gt 0$,当$x\\in U(x_0,\\sigma)$时，有$|f(x)-f(x_0)|\\lt \\epsilon$,则称$f(x)$在$x_0$处连续。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9759db4d-d612-4c62-9ada-bc0195ff793f",
   "metadata": {},
   "source": [
    "**【间断点】**\n",
    "\n",
    "- 第一类间断点:设$x_0$是函数$f(x)$的一个间断点，如果$f(x)$在点$x_0$的左右极限都存在，那么成$x_0$为$f(x)$的第一类间断点。\n",
    "    - 可去间断点: $\\lim\\limits_{x\\rightarrow x_0^-}f(x)=\\lim\\limits_{x\\rightarrow x_0^+}f(x)\\neq\\lim\\limits_{x\\rightarrow x_0}f(x)$\n",
    "    - 跳跃间断点: $\\lim\\limits_{x\\rightarrow x_0^-}f(x)\\neq\\lim\\limits_{x\\rightarrow x_0^+}f(x)$\n",
    "- 第二类间断点：如果$f(x)$在点$x_0$的左右极限至少有一个不存在，那么称$x_0$为$f(x)$的第二类间断点。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7832e6cb-038a-40a2-935a-7fd8c540e059",
   "metadata": {},
   "source": [
    "**【介值定理】：** 设函数$f(x)$在$[a,b]$上连续,$f(a)\\neq f(b)$。若$\\mu$是介于$f(a)$与$f(b)$之间的任意实数，则至少存在一点$x_0\\in(a,b)$,使得$f(x_0) = \\mu$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "09ccb93d-f4ac-4aa0-b9c4-02a83a876218",
   "metadata": {},
   "source": [
    "(1)已知函数$f(x)$在$[a,b]$上连续，且$a<c<d<b$,$f(c)+f(d)=k$,证明：至少存在一点$\\xi\\in(a,b)$使得$2f(\\xi)=k$。\n",
    "\n",
    "解答思路：\n",
    "- $f(c)=f(d)$则$\\xi$为$c$或$d$任意一个值都可以；\n",
    "- $f(c)\\neq f(d)$不妨设$f(c)<f(d)$，使用介值定理。\n",
    "\n",
    "P72"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "47de9dc8-f18d-46a3-b104-aef7767b906d",
   "metadata": {},
   "source": [
    "**【零点存在定理】：** 若函数$f(x)$在$[a,b]$上连续，且$f(a)f(b)\\lt 0$,则至少存在一点$x_0\\in(a,b)$，使得$f(x_0)=0$。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "60fb9645-0fc6-4713-b04f-299a375c0751",
   "metadata": {},
   "source": [
    "(2)在平面有界区域内，由连续曲线$C$围成一个封闭图形。证明：存在实数$\\xi$,使直线$y=x+\\xi$平分该图形的面积。\n",
    "\n",
    "解题思路：设直线簇$y=x+\\zeta$与封闭图形相交，并将其分为面积为$S_1(\\zeta)$和$S_2(\\zeta)$，构造$F(\\zeta)=S_1(\\zeta)-S_2(\\zeta)$为连续函数，使其为0，也就是实现了平分。\n",
    "\n",
    "P72"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2eb78b84-0029-4eca-a15d-ff43061d8405",
   "metadata": {},
   "source": [
    "**【一致连续】：** 设$f(x)$是定义在区间$I$上的函数。若对$\\forall \\epsilon\\gt 0$,$\\exists\\sigma\\gt0$，对$\\forall x_1,x_2\\in I$，只要$\\vert x_1 - x_2\\vert\\lt\\sigma$，就有$$\\vert f(x_1) - f(x_2)\\vert\\lt\\epsilon$$,则称$f(x)$在区间$I$上一致连续。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "848ae745-2e28-4b44-aa34-791eae299088",
   "metadata": {},
   "source": [
    "(3)证明:对$\\forall a\\in (0,1)$,函数$f(x) = \\frac{1}{x}$在$[a,1)$上一致连续。\n",
    "\n",
    "解题思路：将具体$f(x)$代入$\\vert f(x_1) - f(x_2)\\vert\\lt\\epsilon$,证明$\\exists \\sigma$使得不等式成立。\n",
    "\n",
    "P73"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0bc87322-bfbf-4244-adcc-85a3475ea6fa",
   "metadata": {},
   "source": [
    "**【不一致连续】：** 设$f(x)$是定义在$I$上的函数。则$f(x)$在区间$I$上不一致连续的充要条件是：存在$\\epsilon_0\\gt0$及数列$\\{x_n^{'}\\} \\subset I,\\{x_n^{''}\\} \\subset I$,满足$\\vert x^{'} - x^{''}\\vert\\rightarrow 0(n\\rightarrow \\infty)$及$$\\vert f(x_n^{'}) - f(x_n^{''})\\vert\\ge\\epsilon_0$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "17578c79-9636-493e-b522-b9fbe36162d2",
   "metadata": {},
   "source": [
    "(4)证明：函数$f(x)=\\frac{1}{x}$在$(0,1)$上不一致连续。\n",
    "\n",
    "解题思路：取$\\epsilon_0=1$,另$x^{'}=\\frac{1}{n},x^{''}=\\frac{1}{n+1}$\n",
    "\n",
    "P73"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "195d1002-d07d-404e-bc95-e30e508a725a",
   "metadata": {},
   "source": [
    "**【定理】** 设函数$f(x)$在$(a,b)$上连续。函数$f(x)$在$(a,b)$上一致连续的充要条件是存在两个有限的单侧极限$\\lim\\limits_{x\\rightarrow a^+}f(x)$和$\\lim\\limits_{x\\rightarrow b^-}f(x)$。特别地，若$f(x)$在<span style=\"text-decoration: red wavy underline;\">有穷区间</span>$I$上无界，则它在$I$上必定不一致连续。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "efbebecc-0f19-4973-a4b7-48816e8497dc",
   "metadata": {},
   "source": [
    "<span style=\"text-decoration: red wavy underline;\">测试</span>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6f9a048-643a-44a3-86eb-eedce40fd74b",
   "metadata": {},
   "source": [
    "(5)证明：函数$f(x)=x+\\sin x$在$(-\\infty,+\\infty)$上一致连续。\n",
    "\n",
    "解题思路：\n",
    "- 一致连续的定义(因为这个是无穷的区间，所以只能用定义去证明)；\n",
    "- 拉格朗日中值定理。\n",
    "\n",
    "P73"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e0878012-21f3-497a-90a8-d1fbcba7de85",
   "metadata": {},
   "source": [
    "**【要点】：** 函数$f(x)$在区间$I$上连续就是在$I$上的每一点都连续，因此这是一个逐点定义的概念，从本质上看，连续性是局部概念。但是$f(x)$在区间$I$上的一致连续性是由$f$和$I$两者共同确定的整体概念。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8cfbd919-d848-46ad-b6ba-da2adab902e1",
   "metadata": {},
   "source": [
    "### 3.一元函数微分学"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7a08ea3d-95b1-4855-a0a7-2fe666c841c0",
   "metadata": {},
   "source": [
    "**基本初等函数求导公式**\n",
    "1. **常数函数**\n",
    "若 $y = C$（$C$ 为常数），则 $y^\\prime = 0$\n",
    "2. **幂函数**\n",
    "若 $y = x^n$（$n$ 为实数），则 $y^\\prime = nx^{n - 1}$\n",
    "3. **指数函数**\n",
    "    - 若 $y = e^x$，则 $y^\\prime = e^x$\n",
    "    - 若 $y = a^x$（$a>0$ 且 $a\\neq1$），则 $y^\\prime = a^x\\ln a$\n",
    "4. **对数函数**\n",
    "    - 若 $y = \\ln x$（$x>0$），则 $y^\\prime = \\frac{1}{x}$\n",
    "    - 若 $y = \\log_a x$（$a>0$ 且 $a\\neq1$，$x>0$），则 $y^\\prime = \\frac{1}{x\\ln a}$\n",
    "5. **三角函数**\n",
    "    - 若 $y = \\sin x$，则 $y^\\prime = \\cos x$\n",
    "    - 若 $y = \\cos x$，则 $y^\\prime = -\\sin x$\n",
    "    - 若 $y = \\tan x$，则 $y^\\prime = \\sec^2 x=\\frac{1}{\\cos^2 x}$\n",
    "    - 若 $y = \\cot x$，则 $y^\\prime = -\\csc^2 x=-\\frac{1}{\\sin^2 x}$\n",
    "    - 若 $y = \\sec x$，则 $y^\\prime = \\sec x\\tan x$\n",
    "    - 若 $y = \\csc x$，则 $y^\\prime = -\\csc x\\cot x$\n",
    "6. **反三角函数**\n",
    "    - 若 $y = \\arcsin x$（$-1 < x < 1$），则 $y^\\prime = \\frac{1}{\\sqrt{1 - x^2}}$\n",
    "    - 若 $y = \\arccos x$（$-1 < x < 1$），则 $y^\\prime = -\\frac{1}{\\sqrt{1 - x^2}}$\n",
    "    - 若 $y = \\arctan x$，则 $y^\\prime = \\frac{1}{1 + x^2}$\n",
    "    - 若 $y = \\text{arccot} x$，则 $y^\\prime = -\\frac{1}{1 + x^2}$\n",
    "\n",
    "**函数求导的四则运算法则**\n",
    "\n",
    "设 $u = u(x)$，$v = v(x)$ 均可导，则：\n",
    "1. $(u + v)^\\prime = u^\\prime + v^\\prime$\n",
    "2. $(u - v)^\\prime = u^\\prime - v^\\prime$\n",
    "3. $(uv)^\\prime = u^\\prime v + uv^\\prime$\n",
    "4. $(\\frac{u}{v})^\\prime = \\frac{u^\\prime v - uv^\\prime}{v^2}$（$v\\neq0$）\n",
    "\n",
    "**复合函数求导法则（链式法则）**\n",
    "\n",
    "若 $y = f(u)$，$u = g(x)$，且 $f(u)$ 和 $g(x)$ 均可导，则 $(f(g(x)))^\\prime = f^\\prime(u)\\cdot g^\\prime(x)$，即 $\\frac{dy}{dx}=\\frac{dy}{du}\\cdot\\frac{du}{dx}$ "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "01ae8ee7-d40a-40c7-962e-7161e6ad6979",
   "metadata": {},
   "source": [
    "**导数的定义**\n",
    "\n",
    "设函数$f(x)$在点$x_0$的某一邻域$U(x_0,\\sigma_0)(\\sigma_0\\gt 0)$内有定义。若极限$$\\lim\\limits_{x\\rightarrow x_0}\\frac{f(x)-f(x_0)}{x-x_0}$$或者$$\\lim\\limits_{\\Delta x\\rightarrow 0}\\frac{f(x+\\Delta x)-f(x)}{\\Delta x}$$存在，则称$f(x)$在点$x_0$处可导，导数为$f^{'}(x_0)$。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "040ada19-0d7b-45fc-a471-557e9e97e424",
   "metadata": {},
   "source": [
    "**【定理】：** 函数$y = f(x)$在点$x_0$处可微的充要条件是函数$f(x)$在点$x_0$处可导。\n",
    "\n",
    "$\\Delta y =f(x_0+\\Delta x) - f(x_0) =f^{'}(x_0)\\Delta x + o(\\Delta x) = dy + o(\\Delta x)$\n",
    "\n",
    "$dy =f^{'}(x_0)\\Delta x = f^{'}(x_0)dx$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ea8fd9b-2414-433c-aef1-55dcdb7e7e40",
   "metadata": {},
   "source": [
    "**可微、可导和连续的关系：** 函数$f(x)$在点$x_0$处可微$\\Leftrightarrow$函数$f(x)$在点$x_0$处可导$\\Rightarrow$函数$f(x)$在点$x_0$处连续。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "99a9421e-84ac-471c-9597-a9ec2e19e110",
   "metadata": {},
   "source": [
    "**【莱布尼茨公式】：** 设函数$f(x)$和$g(x)$都是$n$阶可导，则它们的积函数也是$n$阶可导，且\n",
    "$$[f(x)g(x)]^{(n)} = \\sum\\limits_{k=0}^nC_n^kf^{(n-k)}(x)g^{(k)}(x)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc39d01c-ba72-4b5c-9fd7-a593e1a4022c",
   "metadata": {},
   "source": [
    "**【费马定理】：** 极值点的导数如果存在，则该点处的导数值为0。\n",
    "\n",
    "- 满足$f^{'}(x)=0$的点$x_0$为函数的驻点，驻点不一定是极值点，极值点可能是驻点或者不可导点。\n",
    "\n",
    "- 通过比较$f(x)$在$[a,b]$上所有的驻点、不可导点和端点的函数值来确定$f(x)$在该区间的最大值和最小值。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6fcbc613-1a13-4fe9-8841-787f7b1be24f",
   "metadata": {},
   "source": [
    "**【罗尔定理】：** 设函数$f(x)$在$[a,b]$上连续，$(a,b)$内可导，且$f(a) = f(b)$，则在$(a,b)$内至少存在一点$c$,有$f^{'}(c) = 0$。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ac95ca91-dbdf-4c7b-ae3b-516b9d2be562",
   "metadata": {},
   "source": [
    "(6)设$f(x)$在$[0,1]$上连续，在$(0,1)$内可导，且$f(1)=0$。证明：在$(0,1)$内存在一点$\\xi$,使得$f^{'}(\\xi) = -\\frac{2f(\\xi)}{\\xi}$\n",
    "\n",
    "解题思路：令$F(x)=x^2f(x)$，则$F(x)$在$[0,1]$上连续，在$(0,1)$内可导,且$F(0)=F(1)=0$,根据罗尔定理可知$\\exists \\xi$，使得$F^{'}(\\xi) = 0$。\n",
    "\n",
    "P84"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c8ecc8e-7b79-46e6-8a65-dd2d7676182c",
   "metadata": {},
   "source": [
    "**【拉格朗日中值定理】：**\n",
    "\n",
    "设函数 $y = f(x)$ 满足以下条件：\n",
    " - 在闭区间 $[a,b]$ 上连续；\n",
    " - 在开区间 $(a,b)$ 内可导。\n",
    "\n",
    "那么在开区间 $(a,b)$ 内至少存在一点 $\\xi$（$a < \\xi < b$），使得等式 $f(b)-f(a)=f'(\\xi)(b - a)$ 成立。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "03b60c95-1635-4147-b023-11768d965878",
   "metadata": {},
   "source": [
    "**【柯西中值定理】：** 设函数$f(x)$和$g(x)$在$[a,b]$上连续，$(a,b)$内可导，且$g^{'}(x)\\neq 0$,则$[a,b]$内至少存在一点$c$,有\n",
    "$$\\frac{f^{'}(x)}{g^{'}(x)} = \\frac{f(b)-f(a)}{g(b)-g(a)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f9ac611-1e89-40d8-887e-41aff168b3ec",
   "metadata": {},
   "source": [
    "**【洛必达法则】：** 适用于$\\frac{0}{0}$或者$\\frac{\\infty}{\\infty}$求极限。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bfe3fe82-c298-4d2d-b791-ae04ec6706cd",
   "metadata": {},
   "source": [
    "(7)求极限$\\lim\\limits_{x\\rightarrow 0}(\\frac{1}{\\sin^2 x}-\\frac{1}{x^2})$\n",
    "\n",
    "解题思路：乘法运算可以使用等价无穷小因子替换$x^2\\sin ^2 x\\sim x^4$,后用运用洛必达法则，最终解为$\\frac{1}{3}$。\n",
    "\n",
    "P88"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5bf085e0-a0f8-4fe0-84e5-a65b9a03060d",
   "metadata": {},
   "source": [
    "**泰勒公式**\n",
    "\n",
    "设函数$f(x)$在$x_0$处具有$n(n\\ge 1)$阶导数：\n",
    "\n",
    "- 泰勒多项式:$$P_n(x)=f(x_0) + f^{'}(x_0)(x-x_0) + \\frac{f^{''}(x_0)}{2!}(x-x_0)^2+\\cdots + + \\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$\n",
    "\n",
    "- 泰勒公式：$$f(x)=f(x_0) + f^{'}(x_0)(x-x_0) + \\frac{f^{''}(x_0)}{2!}(x-x_0)^2+\\cdots + + \\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n + R_n(x)$$\n",
    "\n",
    "- 泰勒公式的余项：$$R_n(x)=f(x) - P_n(x)$$\n",
    "\n",
    "- 皮亚诺余项：$$R_n(x)=o((x-x_0)^n)\\quad(x\\rightarrow x_0)$$\n",
    "\n",
    "- 拉格朗日余项：$$R_n(x)=\\frac{f^{(n+1)}(\\xi)}{(n+1)!}(x-x_0)^{n+1}$$带有拉格朗日余项的泰勒展开式，当$n=0$时就是拉格朗日中值定理。\n",
    "\n",
    "- 当$x_0=0$为麦克劳林公式P89"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ab7fd42-7cf1-4790-81cf-f949c4117301",
   "metadata": {},
   "source": [
    "以下是一些常用函数带有皮亚诺余项的**麦克劳林公式**：\n",
    "\n",
    "① **$e^x$的麦克劳林公式**\n",
    "- $e^x = 1 + x + \\frac{x^2}{2!}+\\frac{x^3}{3!}+\\cdots+\\frac{x^n}{n!}+o(x^n)$\n",
    "    \n",
    "② **$\\sin x$的麦克劳林公式**\n",
    "- $\\sin x = x - \\frac{x^3}{3!}+\\frac{x^5}{5!}-\\frac{x^7}{7!}+\\cdots+(-1)^{k}\\frac{x^{2k + 1}}{(2k + 1)!}+o(x^{2k + 1})$\n",
    "    \n",
    "③ **$\\cos x$的麦克劳林公式**\n",
    "- $\\cos x = 1 - \\frac{x^2}{2!}+\\frac{x^4}{4!}-\\frac{x^6}{6!}+\\cdots+(-1)^{k}\\frac{x^{2k}}{(2k)!}+o(x^{2k})$\n",
    "    \n",
    "④ **$\\ln(1 + x)$的麦克劳林公式**\n",
    "- $\\ln(1 + x)=x-\\frac{x^2}{2}+\\frac{x^3}{3}-\\frac{x^4}{4}+\\cdots+(-1)^{n - 1}\\frac{x^n}{n}+o(x^n)$\n",
    "    \n",
    "⑤ **$(1 + x)^{\\alpha}$的麦克劳林公式**\n",
    "- $(1 + x)^{\\alpha}=1+\\alpha x+\\frac{\\alpha(\\alpha - 1)}{2!}x^2+\\frac{\\alpha(\\alpha - 1)(\\alpha - 2)}{3!}x^3+\\cdots+\\frac{\\alpha(\\alpha - 1)\\cdots(\\alpha - n + 1)}{n!}x^n+o(x^n)$\n",
    "    \n",
    "⑥ **$\\arctan x$的麦克劳林公式**\n",
    "- $\\arctan x = x - \\frac{x^3}{3}+\\frac{x^5}{5}-\\frac{x^7}{7}+\\cdots+(-1)^{n}\\frac{x^{2n + 1}}{2n + 1}+o(x^{2n + 1})$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb9601f3-5136-4b6f-91af-358f3054e2bf",
   "metadata": {},
   "source": [
    "**函数的凹凸性**\n",
    "\n",
    "设函数f(x)在区间$I$上有定义。若对$\\forall x_1,x_2\\in I$,$\\forall \\lambda\\in (0,1)$:\n",
    "\n",
    "- 凸函数(下凸)：$$f(\\lambda x_1 + (1-\\lambda)x_2)\\le \\lambda f(x_1) + (1-\\lambda)f(x_2)$$充要条件是$f^{''}(x)\\ge 0,\\forall x\\in I$\n",
    "  \n",
    "- 凹函数(上凸)：$$f(\\lambda x_1 + (1-\\lambda)x_2)\\ge \\lambda f(x_1) + (1-\\lambda)f(x_2)$$充要条件是$f^{''}(x)\\le 0,\\forall x\\in I$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ae7fdf03-e211-438e-9621-56dcc5f88eed",
   "metadata": {},
   "source": [
    "**拐点**\n",
    "\n",
    "- 定义：设函数$f(x)$在点$x_0$的某一邻域内连续。如果$f(x)$在$x_0$左、右两侧的凹凸性正好相反，则称$x_0$是$f(x)$的一个拐点。\n",
    "\n",
    "- 必要条件：$f^{''}(x_0) = 0$\n",
    "\n",
    "- 第一充分条件：$f^{''}(x)$在$x_0$左右两侧的符号相反。\n",
    "\n",
    "- 第二充分条件：$f^{''}(x_0) = 0$且$f^{'''}(x_0) \\neq 0$\n",
    "\n",
    "P92"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a2b925c-66af-4ed6-a028-091de8e6c45e",
   "metadata": {},
   "source": [
    "### 4.一元函数积分学"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f5c6913-c131-43c8-9f26-4ac837eda8c4",
   "metadata": {},
   "source": [
    "一般地：\n",
    "\n",
    "| 被积函数 | 换元 | \n",
    "| ----: | ----: |\n",
    "| $\\sqrt{a^2 - x^2}$ | $x=a\\sin t$| \n",
    "| $\\sqrt{a^2 + x^2}$ | $x=a\\tan t$ | \n",
    "| $\\sqrt{x^2 - a^2}$ | $x=a\\sec t$ | "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b28a8738-8c2e-472e-89b9-00e9f1d0cd73",
   "metadata": {},
   "source": [
    "**常见的三角函数变换包括以下几种：**\n",
    "\n",
    "**诱导公式**\n",
    "  \n",
    "诱导公式可以将任意角的三角函数转化为锐角三角函数，其规律可概括为“奇变偶不变，符号看象限”。\n",
    "例如：\n",
    "- $\\sin(\\alpha + 2k\\pi) = \\sin\\alpha$，$\\cos(\\alpha + 2k\\pi) = \\cos\\alpha$，$k\\in Z$。\n",
    "- $\\sin(\\pi - \\alpha) = \\sin\\alpha$，$\\cos(\\pi - \\alpha) = -\\cos\\alpha$。\n",
    "- $\\sin(\\frac{\\pi}{2} - \\alpha) = \\cos\\alpha$，$\\cos(\\frac{\\pi}{2} - \\alpha) = \\sin\\alpha$。\n",
    "\n",
    "**同角三角函数的基本关系**\n",
    "  \n",
    "- 平方关系：$\\sin^{2}\\alpha + \\cos^{2}\\alpha = 1$，$1 + \\tan^{2}\\alpha = \\sec^{2}\\alpha$，$1 + \\cot^{2}\\alpha = \\csc^{2}\\alpha$。\n",
    "- 商数关系：$\\tan\\alpha = \\frac{\\sin\\alpha}{\\cos\\alpha}$，$\\cot\\alpha = \\frac{\\cos\\alpha}{\\sin\\alpha}$。\n",
    "\n",
    "**两角和与差的三角函数公式**\n",
    "  \n",
    "- $\\sin(\\alpha + \\beta) = \\sin\\alpha\\cos\\beta + \\cos\\alpha\\sin\\beta$。\n",
    "- $\\sin(\\alpha - \\beta) = \\sin\\alpha\\cos\\beta - \\cos\\alpha\\sin\\beta$。\n",
    "- $\\cos(\\alpha + \\beta) = \\cos\\alpha\\cos\\beta - \\sin\\alpha\\sin\\beta$。\n",
    "- $\\cos(\\alpha - \\beta) = \\cos\\alpha\\cos\\beta + \\sin\\alpha\\sin\\beta$。\n",
    "- $\\tan(\\alpha + \\beta) = \\frac{\\tan\\alpha + \\tan\\beta}{1 - \\tan\\alpha\\tan\\beta}$。\n",
    "- $\\tan(\\alpha - \\beta) = \\frac{\\tan\\alpha - \\tan\\beta}{1 + \\tan\\alpha\\tan\\beta}$。\n",
    "\n",
    "**二倍角公式**\n",
    "  \n",
    "- $\\sin2\\alpha = 2\\sin\\alpha\\cos\\alpha$。\n",
    "- $\\cos2\\alpha = \\cos^{2}\\alpha - \\sin^{2}\\alpha = 2\\cos^{2}\\alpha - 1 = 1 - 2\\sin^{2}\\alpha$。\n",
    "- $\\tan2\\alpha = \\frac{2\\tan\\alpha}{1 - \\tan^{2}\\alpha}$。\n",
    "\n",
    "**半角公式**\n",
    "  \n",
    "- $\\sin\\frac{\\alpha}{2} = \\pm\\sqrt{\\frac{1 - \\cos\\alpha}{2}}$。\n",
    "- $\\cos\\frac{\\alpha}{2} = \\pm\\sqrt{\\frac{1 + \\cos\\alpha}{2}}$。\n",
    "- $\\tan\\frac{\\alpha}{2} = \\pm\\sqrt{\\frac{1 - \\cos\\alpha}{1 + \\cos\\alpha}} = \\frac{\\sin\\alpha}{1 + \\cos\\alpha} = \\frac{1 - \\cos\\alpha}{\\sin\\alpha}$。\n",
    "\n",
    "**辅助角公式**\n",
    "    \n",
    "$a\\sin\\alpha + b\\cos\\alpha = \\sqrt{a^{2} + b^{2}}\\sin(\\alpha + \\varphi)$，其中$\\tan\\varphi = \\frac{b}{a}$。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4731fdbd-3d25-4ca0-9417-97bee546eb12",
   "metadata": {},
   "source": [
    "**分部积分**\n",
    "\n",
    "用来处理<span style=\"text-decoration: blue wavy underline;\">不同函数相乘</span>的问题，把被积函数拆成$u(x)v(x)$，并且将$u(x)v^{'}(x)$ 改写成 $u(x)dv(x)$,按照“反对幂三指”，一般将排序次序在后面的函数作为$v^{'}(x)$优先与$dx$结合成$dv(x)$\n",
    "$$\\int u(x)v^{'}(x)dx = u(x)v(x) - \\int u^{'}(x)v(x)dx$$或\n",
    "$$\\int u(x)dv(x) = u(x)v(x) - \\int v(x)du(x)$$\n",
    "\n",
    "【例】计算不定积分$$\\int e^{2x}\\sin x dx$$\n",
    "解题思路：进行两次分部积分，最终结果$\\frac{1}{5}e^{2x}(2\\sin x + \\cos x) + C$ \n",
    "\n",
    "P97"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19c19e81-fce5-4599-85cd-2acefc92be3e",
   "metadata": {},
   "source": [
    "**积分中值定理**\n",
    "\n",
    "设函数$f(x)$在$[a,b]$上连续，则至少存在一点$c\\in[a,b]$，使得$$\\int_a^b f(x)dx = f(c)(b-a)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d328bcb3-b1ef-414d-b48f-41799727e4da",
   "metadata": {},
   "source": [
    "(1)计算定积分$\\int_0^2\\sqrt{2x-x^2}dx$\n",
    "\n",
    "解题思路：\n",
    "令$y=\\sqrt{2x-x^2}$则有$y^2 + (x-1)^2 =1$，上述定积分表示为半圆面积$\\frac{\\pi}{2}$。\n",
    "\n",
    "P99"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4b17baf8-3d19-4d0c-8adf-ee25db8d951f",
   "metadata": {},
   "source": [
    "**可积的条件**\n",
    "\n",
    "- 必要条件：若$f(x)$在$[a,b]$上可积，则$f(x)$在$[a,b]$上有界。\n",
    "- 充分条件：\n",
    "    - 若$f(x)$在$[a,b]$上连续,则$f(x)$在$[a,b]$上可积；\n",
    "    - 若$f(x)$在$[a,b]$上有界,且只有有限个间断点,则$f(x)$在$[a,b]$上可积；\n",
    "    - 若$f(x)$在$[a,b]$上单调,则$f(x)$在$[a,b]$上可积。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fc7888d9-63b3-462a-9e91-1030090429e1",
   "metadata": {},
   "source": [
    "根据链式法则，若$f(x)$是$[a,b]$上的连续函数，$u(x)$在$[\\alpha,\\beta]$上可导，且对$\\forall x \\in [\\alpha,\\beta]$,有$u(x)\\in[a,b]$,则$[\\int^{u(x)}_a f(t)dt]^{'} = f(u(x))u^{'}(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "95ddb38d-885a-4994-a209-099a68a98393",
   "metadata": {},
   "source": [
    "(2)已知$f(x)$是在$[a,b]$上的连续函数，设$F(x)=\\int^x_a f(t)dt,x\\in[a,b]$,证明：\n",
    "- ①$F(x)$在$[a,b]$上连续；\n",
    "- ②$F(x)$在$[a,b]$上可导，且$F^{'}(x) = f(x)$\n",
    "\n",
    "解题思路：①$f(x)$有界，其定积分也是有界的，利用函数连续的定义；②\n",
    "\n",
    "P102 "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dbc37bc7-4988-4287-a5a4-907fae79a716",
   "metadata": {},
   "source": [
    "(3)利用定积分求极限：$$\\lim\\limits_{n\\rightarrow\\infty}\\frac{1}{n^4}(1+2^3+3^3 + ...n^3)$$\n",
    "解题思路：原式可化为：$\\lim\\limits_{n\\rightarrow\\infty}\\frac{1}{n}((\\frac{1}{n})^3+(\\frac{2}{n})^3+\\cdots+(\\frac{n}{n})^3)=\\lim\\limits_{n\\rightarrow\\infty}\\frac{1}{n}\\sum\\limits_{i=1}^n(\\frac{i}{n})^3$可以看成是$x^3$在区间$[0,1]$上的一个积分和。\n",
    "\n",
    "P103"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2c39b639-4f6d-47a2-bf4b-56dec920219a",
   "metadata": {},
   "source": [
    "(4)**求体积**\n",
    "\n",
    "【1】求由两个圆柱面$x^2+y^2=a^2$和 $x^2+z^2=a^2$所围立体的体积。$$V=\\int_a^b A(x)dx$$\n",
    "【2】导出圆锥体的体积公式。$$V=\\pi\\int_a^b[f(x)]^2dx$$上述是绕$x$轴旋转。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2b25f959-e1a0-4974-af6c-b837fc7fe9f5",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "**绕$y$轴旋转**\n",
    "\n",
    "- **公式推导**：分两种情况。\n",
    "    - 若已知$x = g(y)$（$y$的取值范围是$[c, d]$），由曲线$x = g(y)$，直线$y = c$，$y = d$以及$y$轴所围成的曲边梯形绕$y$轴旋转一周所得到的旋转体，同样用微元法，体积微元$dV = \\pi [g(y)]^2dy$，则旋转体体积$V = \\pi\\int_{c}^{d}[g(y)]^2dy$。\n",
    "    - 当曲线由$y = f(x)$（$x\\in[a, b]$）给出时，使用柱壳法。在区间$[a, b]$内任取一个小区间$[x, x + dx]$，该小区间上对应的小曲边梯形绕$y$轴旋转所得到的薄壁柱壳的体积近似为$dV = 2\\pi x\\cdot f(x)dx$（其中$2\\pi x$是柱壳的周长，$f(x)$是柱壳的高，$dx$是柱壳的厚度）。\n",
    "对体积微元在区间$[a, b]$上进行积分，可得旋转体体积$V = 2\\pi\\int_{a}^{b}x\\cdot f(x)dx$。\n",
    "\n",
    "以上就是利用定积分求旋转体体积的基本方法和公式，在实际应用中，需要根据具体的曲线方程和旋转轴的情况选择合适的方法进行计算。 "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dcfa10de-74ab-43fe-ac31-3774a135643f",
   "metadata": {},
   "source": [
    "**弧长公式**\n",
    "\n",
    "$$L= \\int_{\\alpha}^{\\beta}\\sqrt{[x^{'}(t)]^2+[y^{'}(t)]^2}dt$$或者$$L= \\int_a^b\\sqrt{1+[f^{'}(x)]^2}dx$$\n",
    "\n",
    "**旋转曲面面积**\n",
    "\n",
    "$$S= 2\\pi\\int_{\\alpha}^{\\beta}y(t)\\sqrt{[x^{'}(t)]^2+[y^{'}(t)]^2}dt$$或者$$S = 2\\pi\\int_a^bf(x)\\sqrt{1+[f^{'}(x)]^2}dx$$\n",
    "注意限定条件$y\\ge 0$。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5da0cc0f-64f8-44c9-8a3a-fe5b4e76a826",
   "metadata": {},
   "source": [
    "### 5.级数"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd526344-ebc6-4c17-ab47-cc154a344a9f",
   "metadata": {},
   "source": [
    " **定义$p$级数**\n",
    "   - $p$级数的一般形式为$\\sum_{n = 1}^{\\infty}\\frac{1}{n^{p}}$，当$p = 2$时，级数为$\\sum_{n = 1}^{\\infty}\\frac{1}{n^{2}}=1+\\frac{1}{2^{2}}+\\frac{1}{3^{2}}+\\frac{1}{4^{2}}+\\cdots+\\frac{1}{n^{2}}+\\cdots$。\n",
    "**利用积分判别法证明其收敛性**\n",
    "   - **积分判别法原理**：设$f(x)$是定义在$[1,+\\infty)$上的单调递减、非负的连续函数，那么正项级数$\\sum_{n = 1}^{\\infty}f(n)$与反常积分$\\int_{1}^{+\\infty}f(x)dx$同时收敛或同时发散。\n",
    "   - **构造函数**：对于级数$\\sum_{n = 1}^{\\infty}\\frac{1}{n^{2}}$，构造函数$f(x)=\\frac{1}{x^{2}}$，$x\\in[1,+\\infty)$。函数$f(x)=\\frac{1}{x^{2}}$在$[1,+\\infty)$上单调递减（对$f(x)$求导，$f^\\prime(x)=-\\frac{2}{x^{3}}<0$，$x\\in(1,+\\infty)$），且$f(x)>0$，满足积分判别法的条件。\n",
    "   - **计算反常积分**：计算反常积分$\\int_{1}^{+\\infty}\\frac{1}{x^{2}}dx$。\n",
    "根据反常积分的计算方法，$\\int_{1}^{+\\infty}\\frac{1}{x^{2}}dx=\\lim_{t\\rightarrow+\\infty}\\int_{1}^{t}x^{-2}dx$。\n",
    "由积分公式$\\int x^{n}dx=\\frac{x^{n + 1}}{n+1}+C(n\\neq - 1)$，对于$\\int x^{-2}dx=-\\frac{1}{x}+C$。\n",
    "则$\\lim_{t\\rightarrow+\\infty}\\int_{1}^{t}x^{-2}dx=\\lim_{t\\rightarrow+\\infty}\\left(-\\frac{1}{x}\\right)\\big|_{1}^{t}=\\lim_{t\\rightarrow+\\infty}\\left(-\\frac{1}{t}+1\\right)=1$。\n",
    "   - **得出结论**：因为反}常积分$\\int_{1}^{+\\infty}\\frac{1}{x^{2}}dx$收敛，根据积分判别法，级数$\\sum_{n = 1}^{\\infty}\\frac{1}{n^{2}}$收敛。\n",
    "\n",
    "**利用比较判别法证明其收敛性**\n",
    "   - **比较判别法原理**：设$\\sum_{n = 1}^{\\infty}a_{n}$和$\\sum_{n = 1}^{\\infty}b_{n}$是两个正项级数，且存在正整数$N$，当$n\\geq N$时，$a_{n}\\leq b_{n}$。\n",
    "   - 若$\\sum_{n = 1}^{\\infty}b_{n}$收敛，则$\\sum_{n = 1}^{\\infty}a_{n}$收敛；\n",
    "   - 若$\\sum_{n = 1}^{\\infty}a_{n}$发散，则$\\sum_{n = 1}^{\\infty}b_{n}$发散。\n",
    "   - **构造比较级数**：已知$\\frac{1}{n^{2}}<\\frac{1}{n(n - 1)}=\\frac{1}{n - 1}-\\frac{1}{n}(n\\geq2)$。\n",
    "   - **分析比较级数的敛散性**：考虑级数$\\sum_{n = 2}^{\\infty}\\left(\\frac{1}{n - 1}-\\frac{1}{n}\\right)$，它的前$n$项和$S_{n}=\\left(1-\\frac{1}{2}\\right)+\\left(\\frac{1}{2}-\\frac{1}{3}\\right)+\\cdots+\\left(\\frac{1}{n - 1}-\\frac{1}{n}\\right)=1-\\frac{1}{n}$。\n",
    "当$n\\rightarrow\\infty$时，$\\lim_{n\\rightarrow\\infty}S_{n}=\\lim_{n\\rightarrow\\infty}\\left(1-\\frac{1}{n}\\right)=1$，所以级数$\\sum_{n = 2}^{\\infty}\\left(\\frac{1}{n - 1}-\\frac{1}{n}\\right)$收敛。\n",
    "   - **得出结论**：因为$\\frac{1}{n^{2}}<\\frac{1}{n(n - 1)}(n\\geq2)$，且$\\sum_{n = 2}^{\\infty}\\left(\\frac{1}{n - 1}-\\frac{1}{n}\\right)$收敛，根据比较判别法，级数$\\sum_{n = 1}^{\\infty}\\frac{1}{n^{2}}$收敛（$\\sum_{n = 1}^{\\infty}\\frac{1}{n^{2}} = 1+\\sum_{n = 2}^{\\infty}\\frac{1}{n^{2}}$，前面加一项常数$1$不影响敛散性）。\n",
    "\n",
    "综上，$p$级数当$p = 2$时收敛。 "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "01d1b78a-a6be-41c5-bccb-8328677a7a24",
   "metadata": {},
   "source": [
    "**莱布尼茨判别法**\n",
    "\n",
    "**定理** 若交错级数$\\sum\\limits_{n=1}^{\\infty}(-1)^{n-1}u_n$满足以下两个条件：\n",
    "\n",
    "- 数列$\\{u_n\\}$单调递减；\n",
    "- $\\lim\\limits_{n\\rightarrow\\infty} u_n = 0$\n",
    "\n",
    "则级数$\\sum\\limits_{n=1}^{\\infty}(-1)^{n-1}u_n$收敛。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2ab9b982-80a6-4efb-8f3c-6358470cba73",
   "metadata": {},
   "source": [
    "**收敛半径及收敛域**\n",
    "\n",
    "①使用比值法求幂级数的收敛半径。若$\\lim\\limits_{n\\rightarrow\\infty}\\vert\\frac{a_{n+1}}{a_{n}}\\vert = l$(或者$\\lim\\limits_{n\\rightarrow\\infty}\\sqrt{\\vert a_n\\vert}= l$),则\n",
    "$$R = \\begin{cases}\n",
    "\\frac{1}{l}, & 0 \\lt l \\lt +\\infty \\\\\n",
    "0, & l = +\\infty \\\\\n",
    "+\\infty, & l=0\n",
    "\\end{cases}$$\n",
    "\n",
    "②若$0 \\lt R \\lt +\\infty$,讨论$\\sum\\limits_{n=0}^{\\infty}a_nx^n$在$x=\\pm R$处的敛散性。\n",
    "\n",
    "③写出幂级数的收敛域。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cc1ab0e0-fec1-4fc5-9beb-e70983645e41",
   "metadata": {},
   "source": [
    "设函数$f(x)$在$(a,b)$内连续，则$f(x)$在$(a,b)$内<span style=\"text-decoration: blue wavy underline;\">必存在原函数</span>。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f2ec7121-accd-4f4c-9891-4768990a6f1d",
   "metadata": {},
   "source": [
    "## 线代"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3616021a-9e41-4339-8ebe-f5ce92ca7ad3",
   "metadata": {},
   "source": [
    "### 1.多项式"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5419dc2a-9a2a-4a9d-9c51-e5eafe0c63de",
   "metadata": {},
   "source": [
    "设$f(x)$与$g(x) \\in K[x]$。如果$(f(x),g(x))=1$，那么称$f(x)$与$g(x)$互素。\n",
    "\n",
    "**【定理】：** $K[x]$中两个多项式$f(x)$与$g(x)$互素的充分必要条件是存在$u(x),v(x)\\in K[x]$,使得$$u(x)f(x) + v(x)g(x)=1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bb21b69c-4355-4d11-9388-c04a16778afd",
   "metadata": {},
   "source": [
    "### 2.行列式"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d3d9db22-e4cc-4fe0-bc1b-51a18bfc3f91",
   "metadata": {},
   "source": [
    "(1)计算\n",
    "$$\n",
    "\\begin{vmatrix}\n",
    "1 & 1 & 1 & 1 \\\\\n",
    "2 & 1 & 1 & -3\\\\\n",
    "1 & 2 & 2 & 5 \\\\\n",
    "4 & 3 & 2 & 1\n",
    "\\end{vmatrix}$$\n",
    "解题思路：*n阶行列式D等于它的仍一行(列)的各元素与其对应的代数余子式的乘积之和。* 这里用两次代数余子式，最终答案为1。\n",
    "\n",
    "P136"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b70031d0-1dd2-4e26-a9f8-cb400dac4d0b",
   "metadata": {},
   "source": [
    "(2)证明:n阶范德蒙特行列式\n",
    "$$\n",
    "\\begin{vmatrix}\n",
    "1 & 1 & 1 & \\cdots & 1 \\\\\n",
    "x_1 & x_2 & x_3 & \\cdots  & x_n\\\\\n",
    "x_1^2 & x_2^2 & x_3^2 & \\cdots  & x_n^2 \\\\\n",
    "x_1^3 & x_2^3 & x_3^3 & \\cdots   & x_n^3 \\\\\n",
    ". & . & . &    & . \\\\\n",
    ". & . & . &    & . \\\\\n",
    ". & . & . &    & . \\\\\n",
    "x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & \\cdots   & x_n^{n-1} \\\\\n",
    "\\end{vmatrix} = \\prod\\limits_{1\\le j\\lt i \\le n}(x_i - x_j) $$ \n",
    "\n",
    "解题思路：$n=2$显然成立，把$n$阶范德蒙特行列式第$n-1$行的$-x_1$倍加到$n$行上，然后把第$n-2$行的$-x_1$被加到$n-2$行上，以此类推。\n",
    "\n",
    "P137"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee7e98f9-0044-481b-ae8a-377449366191",
   "metadata": {},
   "source": [
    "**【克拉默法则】：** 系数矩阵的行列式$\\vert A\\vert\\neq 0$,那么非齐次线性方程存在唯一解$(\\frac{|B_1|}{|A|},\\frac{|B_2|}{|A|},\\cdots,\\frac{|B_n|}{|A|})^T$,其中$\\vert B_i\\vert$是把系数矩阵第$i$列换成常数项得到的矩阵的行列式。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3afe0e19-430b-4880-9b8d-9cc488b18616",
   "metadata": {},
   "source": [
    "**行列式的计算**\n",
    "\n",
    "若$A$为$n$阶矩阵，根据矩阵行列式的性质，$\\vert kA\\vert = k^n\\vert A\\vert$，所以$3A$的行列式$\\vert 3A\\vert = 3^n\\vert A\\vert$。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "89a8f417-bacb-42f3-88f2-3025db4ed8c5",
   "metadata": {},
   "source": [
    "### 3.线性方程组"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a5dee2d3-9d75-47cb-a6d1-9971b3eddb87",
   "metadata": {},
   "source": [
    "**线性相关和线性无关**\n",
    "\n",
    "$n$个$n$维行(列)向量$\\alpha_1,\\alpha_2,\\cdots,\\alpha_n$线性相关的充要条件是以$\\alpha_1,\\alpha_2,\\cdots,\\alpha_n$为行(列)向量组的矩阵的行列式等于0。线性无相关的充要条件是以$\\alpha_1,\\alpha_2,\\cdots,\\alpha_n$为行(列)向量组的矩阵的行列式不等于0。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7b04a777-8b1b-4f59-8ca6-757e54050f83",
   "metadata": {},
   "source": [
    "**线性方程解**\n",
    "\n",
    "- 数域$K$上的$n$元线性方程组有解的充要条件是它的系数矩阵的秩与增广矩阵的秩相等。\n",
    "- 系数矩阵的秩等于$n$有唯一解，小于$n$有无穷多解。\n",
    "- 齐次线性方程组有非零解时，它的每一个基础解系所含解向量的个数都是$n-r(A)$。其中，$n$是未知数的个数。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "db574400-ec36-4870-b67f-33d922f5e0c7",
   "metadata": {},
   "source": [
    "(1)求齐次线性方程组$$\\begin{cases}\n",
    "x_1 + x_2 - x_3 - x_4=0 \\\\\n",
    "x_1 + 2x_2 + 2x_3 + 3x_4 =0 \\\\\n",
    "2x_1 + 3x_2 + x_3 + 2x_4 =0 \n",
    "\\end{cases}$$的通解。\n",
    "\n",
    "P153"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "018d5a9d-01e1-477a-bbd8-24ffa615dc41",
   "metadata": {},
   "source": [
    "设$\\eta$是$n$元非齐次线性方程组$Ax=b$的一个解(如果有$\\eta_1,\\eta_2,\\cdots$取其中一个作为$\\eta$)，$\\xi$是导出组$Ax=0$的通解，则$\\eta+\\xi$是$n$元非齐次线性方程组的通解。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9805b184-bf14-4fdf-90f7-4eadfb3832b9",
   "metadata": {},
   "source": [
    "(2)求线性方程组$$\\begin{cases}\n",
    "x_1 + x_2 - 2x_4=-6 \\\\\n",
    "4x_1 - x_2 - x_3 - x_4 =1 \\\\\n",
    "3x_1 - x_2 - x_3 =3 \n",
    "\\end{cases}$$的通解。\n",
    "\n",
    "P154"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "366bb87e-6b87-4478-88cb-b1cb767ae21b",
   "metadata": {},
   "source": [
    "### 4.矩阵"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69c6beb2-3298-4933-b32e-7625ea28a31c",
   "metadata": {},
   "source": [
    "$K^n$的非零子空间$V$的一个基所含向量的个数叫做$V$的维数，记作$dimV$,就是矩阵的秩。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "176aab68-17f1-42bb-b127-5c1f3ff019b8",
   "metadata": {},
   "source": [
    "矩阵$A$可逆的充要条件是矩阵$A$的行列式$\\vert A\\vert \\neq 0$。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7d41541d-e90c-469a-a412-d24ee433c56a",
   "metadata": {},
   "source": [
    "矩阵$A$和它的伴随矩阵$A^*$秩的关系与矩阵$A$的秩$r(A)$有关，以下分三种情况进行讨论：\n",
    "\n",
    "**情况一：当$r(A)=n$（$n$为矩阵$A$的阶数）时**\n",
    "\n",
    "- **推理过程**：已知$A A^* = |A|E$，因为$r(A)=n$，所以矩阵$A$可逆，即$|A|\\neq 0$。在等式$A A^* = |A|E$两边同时取行列式可得$|A A^*| = ||A|E|$，根据行列式的性质$|AB| = |A|\\times|B|$以及$|\\lambda E|=\\lambda^n$（$\\lambda$为常数），有$|A|\\times|A^*| = |A|^n$。由于$|A|\\neq 0$，等式两边同时除以$|A|$，得到$|A^*| = |A|^{n - 1}\\neq 0$，这表明矩阵$A^*$可逆，而可逆矩阵的秩等于其阶数，所以$r(A^*) = n$。\n",
    "\n",
    "**情况二：当$r(A)=n - 1$时**\n",
    "\n",
    "- **推理过程**：因为$r(A)=n - 1$，所以$|A| = 0$，则$A A^* = |A|E = O$（零矩阵）。根据矩阵秩的性质：若$AB = O$，则$r(A)+r(B)\\leq n$（$n$为$A$的列数或$B$的行数），在这里$B = A^*$，所以$r(A)+r(A^*)\\leq n$，即$r(A^*) \\leq n - r(A)=n-(n - 1)=1$。又因为$r(A)=n - 1$，说明矩阵$A$至少存在一个$n - 1$阶子式不为零，而伴随矩阵$A^*$的元素是由$A$的$n - 1$阶代数余子式构成的，所以$A^*$中至少有一个元素不为零，即$r(A^*) \\geq 1$。综合可得$r(A^*) = 1$。\n",
    "\n",
    "**情况三：当$r(A)<n - 1$时**\n",
    "\n",
    "- **推理过程**：由于$r(A)<n - 1$，这意味着矩阵$A$的所有$n - 1$阶子式都为零，而伴随矩阵$A^*$的元素是$A$的$n - 1$阶代数余子式，所以$A^*$的所有元素都为零，即$A^* = O$，零矩阵的秩为$0$，所以$r(A^*) = 0$。\n",
    "\n",
    "\n",
    "综上，矩阵$A$和它的伴随矩阵$A^*$秩的关系为$r(A^*)=\\begin{cases}n, & r(A)=n \\\\ 1, & r(A)=n - 1 \\\\ 0, & r(A)<n - 1\\end{cases}$。 "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dfec0a56-8b6e-4d8b-87dc-c97bb6f31299",
   "metadata": {},
   "source": [
    "**伴随矩阵**\n",
    "\n",
    "矩阵$A$和它的伴随矩阵$A^*$满足矩阵乘法的交换律，即$AA^* = A^*A$。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d41e1eb6-18e1-4514-a0d7-43753c37c853",
   "metadata": {},
   "source": [
    "**注意！！！**\n",
    "设$$A =\\begin{bmatrix}a&b\\\\c&d\\end{bmatrix}$$则伴随矩阵$A*$为\n",
    "$$A* =\\begin{bmatrix}d&-b\\\\-c&a\\end{bmatrix}$$\n",
    "注意伴随矩阵的定义。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f2c4fa9-d688-4ef8-b84c-98871c5e28e7",
   "metadata": {},
   "source": [
    "**初等矩阵**\n",
    "\n",
    "**定义：** 对单位矩阵$E$实施一次行(列)变换得到的矩阵称为初等矩阵。\n",
    "\n",
    "在矩阵$A$左边乘一个初等矩阵相当于对$A$做相应的行变换；在矩阵$A$右边乘一个初等矩阵等于对A做相应的列变换。需要注意第三类初等矩阵$E(i,j(k))$,在矩阵$A$左边乘$E(i,j(k))$是将矩阵第j行的k倍加到第i行，右边乘$E(i,j(k))$将矩阵第i列的k倍加到第j列。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "91fa2107-d2fd-494d-961b-a154150924e1",
   "metadata": {},
   "source": [
    "(1)设$A$是3阶方阵，将$A$的第一列和第二列交换得到$B$，在把$B$的第二列加到第三列得$C$,则满足$AQ=C$的可逆矩阵$Q$是$$Q =\\begin{bmatrix}0&1&1\\\\1&0&0\\\\0&0&1\\end{bmatrix}$$解题思路：一个矩阵右乘初等矩阵的进行初等列变换。\n",
    "\n",
    "P162"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3d46b6b-ae9b-40f7-9bc4-3197bf82a817",
   "metadata": {},
   "source": [
    "所有初等矩阵都是可逆的，并且$$E(i,j)^{-1} = E(i,j),E(i(k))^{-1} = E(i(k^{-1})),E(i,j(k))^{-1} = E(i,j(-k))$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f17fb95a-14f4-40dc-b4cd-e35105decc64",
   "metadata": {},
   "source": [
    "矩阵$A$可逆的充要条件是它能表示成有限个初等矩阵的乘积，即$A=P_1P_2\\cdots P_m$,其中$P_1,P_2,\\cdots,P_m$均为初等矩阵。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3e30d0f-77c8-4f8c-aa81-f80d51562101",
   "metadata": {},
   "source": [
    "(2)求矩阵$$A =\\begin{bmatrix}1&1&1\\\\0&1&1\\\\0&0&1\\end{bmatrix}$$的逆矩阵。\n",
    "\n",
    "解题思路：对矩阵(A,E)做初等行变换，把矩阵$A$化为单位矩阵，此时$E$就成为$A$的逆矩阵$A^{-1}$。\n",
    "\n",
    "P163"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7bbced4e-2ef8-4f91-83fb-a9b6e24e9773",
   "metadata": {},
   "source": [
    "求一个可逆矩阵的逆矩阵有两种常用方法：\n",
    "\n",
    "- 伴随矩阵法(适用于阶数不超过3的低阶矩阵):$A^{-1} = \\frac{A*}{\\vert A\\vert}$;\n",
    "- 初级行变换法：将$(A,E)化为(E,A^{-1})$。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b99b9c9-f107-48dd-9b88-c6cae6743b67",
   "metadata": {},
   "source": [
    "(3)在线性空间$R^3$中，已知向量$\\alpha_1=(1,2,1),\\alpha_2=(2,1,4),\\alpha_3=(0,-3,2)$，记$V_1=\\{\\lambda\\alpha_1+\\mu\\alpha_2|\\lambda,\\mu\\in R\\},V_2=\\{k\\alpha_3|k\\in R\\}$。另$V_3=\\{t_1\\eta_1+t_2\\eta_2|t_1,t_2\\in R,\\eta_1\\in V_1,\\eta_2\\in V_2\\}$。\n",
    "\n",
    "①求子空间$V_3$的维数。\n",
    "\n",
    "②求子空间$V_3$的一个标准正交基。\n",
    "\n",
    "解题思路：①$dimV_3 = r(\\alpha_1,\\alpha_2,\\alpha_3)$；②施密特正交化及向量单位化。\n",
    "\n",
    "P165"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4e7d4796-9f3e-4da7-9988-a0be1ab95ded",
   "metadata": {},
   "source": [
    "**常见的线性变换**\n",
    "\n",
    "**旋转变换**\n",
    "\n",
    "旋转变换对应的矩阵为$$\\begin{bmatrix}\\cos\\theta&-\\sin\\theta\\\\ \\sin\\theta&\\cos\\theta \\end{bmatrix}$$,该变换将平面内任意一点(向量)或图形绕原点逆时针旋转$\\theta$角。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "49ad5bb8-80bc-44c0-957b-3591fbd50c0b",
   "metadata": {},
   "source": [
    "平面上的恒等、平移、旋转、反射变换都是保距变换。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e5aa895d-9fa4-437f-9fa5-6d542c2e6d19",
   "metadata": {},
   "source": [
    "### 5.特征值特征向量"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6035d2df-c09a-4557-a354-d88bf60b1630",
   "metadata": {},
   "source": [
    "(1)若矩阵$$A =\\begin{bmatrix}1&-1&1\\\\x&4&y\\\\-3&-3&5\\end{bmatrix}$$有三个线性无关的特征向量,$\\lambda=2$是$A$的二重特征根,求$x,y$的值。\n",
    "\n",
    "解题思路：特征值$\\lambda=2$时二重特征根对应两个线性无关的特征向量，意味着齐次线性方程$(2E-A)x=0$有两组基础解，即$n-r(2E-A) =2,r(2E-A)=1$。最终可以求得$x=2,y=-2$。\n",
    "\n",
    "P172"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "97f1a0b6-f9f6-4aab-b4de-40f72f9e4eb2",
   "metadata": {},
   "source": [
    "全部特征值的和等于矩阵的迹(即主对角线元素的和)tr(A),全部特征值的积等于矩阵A的行列式$\\vert A\\vert$。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "27864919-f283-4d1b-9838-5fa4ee1bc268",
   "metadata": {},
   "source": [
    "**矩阵相似**\n",
    "\n",
    "**定义:** 设$A,B$是数域K上的两个$n$阶矩阵。如果存在数域$K$上n阶可逆矩阵$P$,使得$B = P^{-1}AP$，那么成$A$与$B$相似，记作$A\\sim B$。\n",
    "\n",
    "**扩展:** 如果两个矩阵相似，那么他们具有相同的行列式、相同的秩、相同的迹、相同的特征值。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e2974d1-f352-441c-a9c7-b5ce08ef54e8",
   "metadata": {},
   "source": [
    "(2)求矩阵$$A =\\begin{bmatrix}1&1&-1\\\\-2&4&-2\\\\-2&-2&0\\end{bmatrix}$$的特征值，并判断矩阵$A$是否可对角化，如果矩阵$A$可对角化，求出可逆矩阵$P$,使得$P^{-1}AP$为对角矩阵。\n",
    "\n",
    "解题思路：求出特征值后，求特征向量，有3组线性无关的特征向量，所以矩阵A可以对角化，可以矩阵就是$(\\alpha_1,\\alpha_2,\\alpha_3)$\n",
    "\n",
    "P174"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0b60893b-673d-4e5d-a790-63f8abf4fff2",
   "metadata": {},
   "source": [
    "对于一般的矩阵而言，不同特征值对应的特征向量不一定正交。如果矩阵$A$是实对称矩阵，那么不同特征值对应的特征向量一定正交。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e01a074e-659f-42f8-b9ab-4299504aaa52",
   "metadata": {},
   "source": [
    "求正交矩阵$T$,使得$T^{-1}AT$为对角矩阵，并且写出这个对角矩阵，其中$$A =\\begin{bmatrix}0&-1&-1\\\\-1&0&1\\\\1&1&0\\end{bmatrix}$$\n",
    "解题思路：求出特征值，然后进行施密特正交化和单位化。特征值为1(二重)对应的基础解析为$\\alpha_1 = [1,0,1]^T,\\alpha_2 = [0,1,1]^T$,特征值2对应的基础解系$\\alpha_3 = [1,1,-1]^T$,最终$$T =\\begin{bmatrix}\\frac{\\sqrt{2}}{2}&-\\frac{\\sqrt{6}}{6}&\\frac{\\sqrt{3}}{3}\n",
    "\\\\0&\\frac{\\sqrt{6}}{3}&\\frac{\\sqrt{3}}{3}\\\\\\frac{\\sqrt{2}}{2}&\\frac{\\sqrt{6}}{6}&-\\frac{\\sqrt{3}}{3}\\end{bmatrix}$$\n",
    "\n",
    "P175"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14577868-8879-4ba8-b713-9539953b35fe",
   "metadata": {},
   "source": [
    "**【注意以下容易发生错误的地方】：** \n",
    "- 一般地，对矩阵进行列变换可能会改变矩阵的行空间，而最简梯形矩阵是由矩阵的行空间决定的，所以将矩阵的所有列加到第一列，通常会使最简梯形矩阵发生变化。\n",
    "- 矩阵所有列的值加到第一列，那么行列式不会发生变化。\n",
    "- 可以通过行变换进行最简梯形矩阵化简，但是行变换会改变行列式的符号。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f7cf831-7676-44f0-9874-dcec35867a49",
   "metadata": {},
   "source": [
    "### 6.二次型"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b0f1062c-67d9-4da4-b79a-d9ec80e3bda2",
   "metadata": {},
   "source": [
    "- 二次型和它的矩阵是唯一决定的，且二次型的矩阵都是对称矩阵；\n",
    "- 两个二次型等价的充要条件是它们的矩阵合同；\n",
    "- 如果二次型$f(x_1,x_2,\\cdots,x_n)$等价于一个只含平方项的二次型$g(y_1,y_2,\\cdots,y_n)$，那么称这二次型$g(y_1,y_2,\\cdots,y_n)$是二次型$f(x_1,x_2,\\cdots,x_n)$的**标准型**。\n",
    "- 化二次型$f(x_1,x_2,\\cdots,x_n)=x^TAx$为标准型，就是寻找可逆矩阵$P$,使$P^TAP$为对角阵。\n",
    "- 设二次型$g(y_1,y_2,\\cdots,y_n)$是二次型$f(x_1,x_2,\\cdots,x_n)$的标准型。如果$g(y_1,y_2,\\cdots,y_n)$中平方项的系数仅为1，-1，或0，且系数为1的平方项都在前面，即$g(y_1,y_2,\\cdots,y_n)=y_1^2 + \\cdots+y_p^2 - \\cdots - y_r^2$，称二次型$g(y_1,y_2,\\cdots,y_n)$是二次型$f(x_1,x_2,\\cdots,x_n)$的**规范型**。(这里的$r$就是二次型的秩)\n",
    "- 两个$n$元实二次型等价的充要条件是它们的正惯性指数和负惯性指数都相同。\n",
    "- 设有$n$元实二次型$x^TAx$。如果对$\\underset{\\cdots\\cdots}{\\forall x\\neq 0}$,$x\\in R^n$,都有$x^TAx \\gt 0$,那么称二次型$x^TAx$是正定的，并称矩阵$A$是正定矩阵。\n",
    "- 二次型$x^TAx$是正定二次型的充要条件是矩阵$A$的顺序主子式全大于零。以下结论与该定理等价：\n",
    "    - ①对于任意非零的$n$维列向量$x$,$x^TAx\\gt 0$;\n",
    "    - ②二次型$x^TAx$的正惯性指数等于$n$；\n",
    "    - ③矩阵$A$的特征值全大于0；\n",
    "    - ④矩阵$A$与单位矩阵$E$合同；\n",
    "    - ⑤存在可逆矩阵$Q$,使得$A=Q^TQ$。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "081ef4e4-3c26-4002-be8c-1e23c34c016b",
   "metadata": {},
   "source": [
    "## 空间解析几何"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab156200-c921-411f-87a6-6fec9718c708",
   "metadata": {},
   "source": [
    "### 1.仿射坐标系与向量的外积和混合积"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5d7a4361-bd53-4ca6-93cc-f5e0a9c630d7",
   "metadata": {},
   "source": [
    "两个向量$\\overrightarrow a,\\overrightarrow b$的外积是一个向量，方向是同时垂直于$\\overrightarrow a,\\overrightarrow b$,规定长度为$| \\overrightarrow a\\times  \\overrightarrow b| = |\\overrightarrow a| |\\overrightarrow b|\\sin<\\overrightarrow a,\\overrightarrow b>$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ffb30be1-7bb1-4466-8fe3-30a8828bb033",
   "metadata": {},
   "source": [
    "- 向量**a,b,c**共面的充要条件是(**a,b,c**)=0;\n",
    "- 向量**a,b,c**不共面时，混合积(**a,b,c**)等于以**a,b,c**为棱的平面六面体的有向体积。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4e45e458-21e9-4c10-909f-a4cae2493ace",
   "metadata": {},
   "source": [
    "### 2.空间的平面与直线"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d3870c20-39a0-4302-aeb9-ef797684537f",
   "metadata": {},
   "source": [
    "要理解为什么直线的方向向量可以由两个相交平面的法向量外积所得，需要从平面法向量和直线方向向量的定义以及向量外积的性质来分析，以下是具体解释：\n",
    "\n",
    "**相关定义**\n",
    "  |\n",
    "- **平面法向量**：对于一个平面，其法向量是与该平面垂直的非零向量。若有平面$\\alpha$，其法向量为$\\vec{n_1}$，那么$\\vec{n_1}$与平面$\\alpha$内的任意向量都垂直。同理，对于平面$\\beta$，其法向量$\\vec{n_2}$与平面$\\beta$内的任意向量也垂直。\n",
    "- **直线方向向量**：直线的方向向量是指和这条直线平行或共线的非零向量。对于两个相交平面$\\alpha$和$\\beta$，它们的交线$l$的方向向量$\\vec{v}$与交线$l$是平行关系。\n",
    "\n",
    "**向量外积性质**\n",
    "\n",
    "- **定义**：两个向量$\\vec{a}$和$\\vec{b}$的外积$\\vec{a}\\times\\vec{b}$是一个向量，它的模$\\vert\\vec{a}\\times\\vec{b}\\vert=\\vert\\vec{a}\\vert\\vert\\vec{b}\\vert\\sin\\theta$，其中$\\theta$为$\\vec{a}$与$\\vec{b}$的夹角，它的方向垂直于$\\vec{a}$和$\\vec{b}$所确定的平面，且符合右手定则。\n",
    "- **垂直性**：根据外积的定义，$\\vec{n_1}\\times\\vec{n_2}$所得向量同时垂直于$\\vec{n_1}$和$\\vec{n_2}$。\n",
    "\n",
    "**推导过程**\n",
    "\n",
    "因为平面$\\alpha$的法向量$\\vec{n_1}$垂直于平面$\\alpha$，平面$\\beta$的法向量$\\vec{n_2}$垂直于平面$\\beta$，而交线$l$既在平面$\\alpha$内又在平面$\\beta$内，所以交线$l$的方向向量$\\vec{v}$既垂直于$\\vec{n_1}$又垂直于$\\vec{n_2}$。又因为$\\vec{n_1}\\times\\vec{n_2}$这个向量也同时垂直于$\\vec{n_1}$和$\\vec{n_2}$，所以$\\vec{n_1}\\times\\vec{n_2}$与$\\vec{v}$是平行或共线关系，即直线$l$的方向向量$\\vec{v}$可以由两个相交平面的法向量$\\vec{n_1}$和$\\vec{n_2}$的外积所得。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f513533-8244-43f3-a44b-8dfd4581b051",
   "metadata": {},
   "source": [
    "(1)在空间直角坐标系下，试判断直线$l:\\begin{cases}\n",
    "2x + y + z - 1 = 0 \\\\\n",
    "x + 2y -z -2 =0\n",
    "\\end{cases}$与平面$\\pi:3x - y + 2z + 1 = 0$的位置关系，并求出直线$l$与平面$\\pi$的夹角的正弦值。\n",
    "\n",
    "解题思路：①求出直线$l$的方向向量**m**；②计算**m**与平面$\\pi$的法向量**n**的余弦值的绝对值，就是直线$l$与平面$\\pi$的夹角的正弦值。\n",
    "\n",
    "P198"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8c913548-abdd-4c0b-80fc-6ab1a2011cf8",
   "metadata": {},
   "source": [
    "#### 空间中点到直线的距离公式由来\n",
    "\n",
    "**向量法推导思路**\n",
    "  \n",
    "设空间中直线 $l$ 过点 $M_0(x_0,y_0,z_0)$，直线的方向向量为 $\\vec{s}=(m,n,p)$，空间中另有一点 $M(x_1,y_1,z_1)$，要求点 $M$ 到直线 $l$ 的距离 $d$。\n",
    "\n",
    "**具体推导过程**\n",
    "\n",
    "1. 首先，构造向量 $\\overrightarrow{M_0M}=(x_1 - x_0,y_1 - y_0,z_1 - z_0)$。\n",
    "2. 根据向量的几何意义，点 $M$ 到直线 $l$ 的距离 $d$、向量 $\\overrightarrow{M_0M}$ 以及向量 $\\vec{s}$ 构成一个以 $\\overrightarrow{M_0M}$ 为斜边的直角三角形（其中 $d$ 是直角边）。\n",
    "   - 我们知道三角形面积 $S$ 有两种表示方法。一方面，根据向量叉积的几何意义，以向量 $\\overrightarrow{M_0M}$ 和 $\\vec{s}$ 为邻边的平行四边形的面积 $S = \\vert\\overrightarrow{M_0M}\\times\\vec{s}\\vert$，而点 $M$ 到直线 $l$ 的距离 $d$ 对应的是以 $\\vec{s}$ 为底，$d$ 为高的三角形面积的两倍与 $\\vec{s}$ 的模的关系，即平行四边形面积 $S=\\vert\\vec{s}\\vert\\cdot d$。\n",
    "3. 然后求解距离 $d$：\n",
    "   - 由 $S = \\vert\\overrightarrow{M_0M}\\times\\vec{s}\\vert=\\vert\\vec{s}\\vert\\cdot d$，可得点 $M$ 到直线 $l$ 的距离公式 $d=\\frac{\\vert\\overrightarrow{M_0M}\\times\\vec{s}\\vert}{\\vert\\vec{s}\\vert}$。\n",
    "   - 若将向量坐标代入，$\\overrightarrow{M_0M}=(x_1 - x_0,y_1 - y_0,z_1 - z_0)$，$\\vec{s}=(m,n,p)$，$\\overrightarrow{M_0M}\\times\\vec{s}=\\left|\\begin{array}{ccc}\\vec{i}&\\vec{j}&\\vec{k}\\\\x_1 - x_0&y_1 - y_0&z_1 - z_0\\\\m&n&p\\end{array}\\right|=\\vec{i}\\left[(y_1 - y_0)p-(z_1 - z_0)n\\right]-\\vec{j}\\left[(x_1 - x_0)p-(z_1 - z_0)m\\right]+\\vec{k}\\left[(x_1 - x_0)n-(y_1 - y_0)m\\right]$，$\\vert\\overrightarrow{M_0M}\\times\\vec{s}\\vert=\\sqrt{[(y_1 - y_0)p-(z_1 - z_0)n]^2+[(x_1 - x_0)p-(z_1 - z_0)m]^2+[(x_1 - x_0)n-(y_1 - y_0)m]^2}$，$\\vert\\vec{s}\\vert=\\sqrt{m^{2}+n^{2}+p^{2}}$。\n",
    "\n",
    "\n",
    "#### 空间中点到平面的距离公式由来\n",
    "\n",
    "**向量法推导思路**\n",
    "\n",
    "设平面 $\\Pi$ 的方程为 $Ax + By+ Cz+D = 0$，其中 $\\vec{n}=(A,B,C)$ 是平面的法向量，空间中一点 $P(x_0,y_0,z_0)$，要求点 $P$ 到平面 $\\Pi$ 的距离 $d$。\n",
    "\n",
    "**具体推导过程**\n",
    "\n",
    "1. 在平面 $\\Pi$ 上任取一点 $Q(x_1,y_1,z_1)$，则有 $Ax_1 + By_1 + Cz_1+D = 0$。\n",
    "2. 构造向量 $\\overrightarrow{QP}=(x_0 - x_1,y_0 - y_1,z_0 - z_1)$。\n",
    "3. 点 $P$ 到平面 $\\Pi$ 的距离 $d$ 就是向量 $\\overrightarrow{QP}$ 在平面法向量 $\\vec{n}$ 上投影的绝对值。\n",
    "   - 根据向量投影公式，向量 $\\overrightarrow{QP}$ 在向量 $\\vec{n}$ 上的投影为 $\\text{Prj}_{\\vec{n}}\\overrightarrow{QP}=\\frac{\\overrightarrow{QP}\\cdot\\vec{n}}{\\vert\\vec{n}\\vert}$。\n",
    "   - 计算 $\\overrightarrow{QP}\\cdot\\vec{n}=A(x_0 - x_1)+B(y_0 - y_1)+C(z_0 - z_1)=Ax_0 + By_0+ Cz_0-(Ax_1 + By_1 + Cz_1)$，因为 $Ax_1 + By_1 + Cz_1=-D$，所以 $\\overrightarrow{QP}\\cdot\\vec{n}=Ax_0 + By_0 + Cz_0+D$。\n",
    "   - 又 $\\vert\\vec{n}\\vert=\\sqrt{A^{2}+B^{2}+C^{2}}$。\n",
    "4. 从而得到点 $P(x_0,y_0,z_0)$ 到平面 $Ax + By + Cz+D = 0$ 的距离公式 $d = \\frac{\\vert Ax_0 + By_0 + Cz_0+D\\vert}{\\sqrt{A^{2}+B^{2}+C^{2}}}$。 "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff4b0212-d7fb-43f1-a24b-70553e1b6880",
   "metadata": {},
   "source": [
    "#### 向量法推导异面直线距离公式\n",
    "\n",
    "**已知条件设定**\n",
    "设异面直线 $l_1$、$l_2$，$l_1$ 过点 $A$，方向向量为 $\\vec{s_1}$；$l_2$ 过点 $B$，方向向量为 $\\vec{s_2}$。两异面直线的公垂向量为 $\\vec{n}$，根据公垂向量的定义，$\\vec{n}$ 同时垂直于 $\\vec{s_1}$ 和 $\\vec{s_2}$，所以可通过向量叉乘得到 $\\vec{n}=\\vec{s_1}\\times\\vec{s_2}$。\n",
    "\n",
    "**推导过程**\n",
    "\n",
    "1. 连接 $A$、$B$ 两点，得到向量 $\\overrightarrow{AB}$。\n",
    "2. 异面直线 $l_1$ 与 $l_2$ 的距离 $d$ 就是向量 $\\overrightarrow{AB}$ 在公垂向量 $\\vec{n}$ 上投影的绝对值。\n",
    "    - 根据向量投影的定义，向量 $\\overrightarrow{AB}$ 在向量 $\\vec{n}$ 上的投影为$\\text{Prj}_{\\vec{n}}\\overrightarrow{AB}=\\frac{\\overrightarrow{AB}\\cdot\\vec{n}}{\\vert\\vec{n}\\vert}$。\n",
    "    - 所以异面直线 $l_1$ 与 $l_2$ 的距离 $d = \\frac{\\vert\\overrightarrow{AB}\\cdot\\vec{n}\\vert}{\\vert\\vec{n}\\vert}$，又因为 $\\vec{n}=\\vec{s_1}\\times\\vec{s_2}$，故距离公式可写为 $d=\\frac{\\vert\\overrightarrow{AB}\\cdot(\\vec{s_1}\\times\\vec{s_2})\\vert}{\\vert\\vec{s_1}\\times\\vec{s_2}\\vert}$。\n",
    "    - 从几何意义上看，$\\vert\\vec{s_1}\\times\\vec{s_2}\\vert$ 是以 $\\vec{s_1}$ 和 $\\vec{s_2}$ 为邻边的平行四边形的面积，$\\overrightarrow{AB}\\cdot(\\vec{s_1}\\times\\vec{s_2})$ 的绝对值表示以 $\\overrightarrow{AB}$、$\\vec{s_1}$ 和 $\\vec{s_2}$ 为棱的平行六面体的体积。而异面直线的距离 $d$ 就相当于以这个平行四边形为底，平行六面体的高。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e9fa5bc6-bb07-4f7a-930d-b729d3b9e394",
   "metadata": {},
   "source": [
    "(2)在空间直角坐标系下，试判断直线$l_1:\\begin{cases}\n",
    "x + y + 1 = 0 \\\\\n",
    "x + 2y + z + 2 =0\n",
    "\\end{cases}$与直线$l_2:\\frac{x-1}{2} = \\frac{y+1}{1} = \\frac{z}{1}$的位置关系，并求这两条直线间的距离。\n",
    "\n",
    "解题思路：①计算$l_1$的方向向量$\\overrightarrow s_1$；②取$y=0$,计算经过$l_1$的某一个点$M_1$;③设$l_2$的方向向量$\\overrightarrow s_2$,经过点$M_2$,计算混合积$(\\overrightarrow s_1,\\overrightarrow s_2,\\overrightarrow{M_1M_2})$，不为0，则异面；④计算异面直线间的距离。\n",
    "\n",
    "P200"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "773cf611-87f6-4d86-a8a1-f93c099ccfe0",
   "metadata": {},
   "source": [
    "(3)求经过直线$l:\\begin{cases}\n",
    "4x - y + 3z - 1 = 0 \\\\\n",
    "x + 5y - z + 2 =0\n",
    "\\end{cases}$和点$(1,2,3)$的平面方程。\n",
    "\n",
    "解题思路：设所求平面方程为$\\lambda(4x - y + 3z - 1) + \\mu(x + 5y - z + 2)=0$;②代入点$(1,2,3)$，求满足方程的$\\lambda$和$\\mu$。\n",
    "\n",
    "P200"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8938d879-1d03-480e-84b1-d3fcb86f3e78",
   "metadata": {},
   "source": [
    "### 3.空间曲面和曲线"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8dbf4b5f-6e35-431a-95e9-d204078898ab",
   "metadata": {},
   "source": [
    "设有$xOy$平面上的一条曲线$\\Gamma :\\begin{cases}\n",
    "F(x,y) = 0 \\\\\n",
    "z =0\n",
    "\\end{cases}$则曲线$\\Gamma$绕$x$轴旋转产生的曲面方程为$$F(x,\\pm\\sqrt{y^2+z^2})=0,$$\n",
    "曲线$\\Gamma$绕$y$轴旋转产生的曲面方程为$$F(\\pm\\sqrt{x^2+z^2},y)=0。$$\n",
    "类似地,可得到$yOz$面或$xOz$面上的曲线绕其所在的坐标旋转产生的旋转曲面方程。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "27e99118-4900-41a3-8357-9b590699cf9b",
   "metadata": {},
   "source": [
    "#### 常见的曲面方程"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3ff48fd6-0dd5-48dc-9ce8-ec8559399e92",
   "metadata": {
    "jp-MarkdownHeadingCollapsed": true
   },
   "source": [
    "\n",
    "以下是常见曲面方程的直角坐标系公式及参数公式整理：\n",
    "\n",
    "\n",
    "**①球面**\n",
    "- **直角坐标系方程**：  \n",
    "  $\n",
    "  (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\n",
    "  $\n",
    "  （球心为 $(a,b,c)$，半径为 $r$）\n",
    "\n",
    "- **参数方程**：  \n",
    "  $\n",
    "  \\begin{cases}\n",
    "  x = a + r \\sin\\theta \\cos\\phi \\\\\n",
    "  y = b + r \\sin\\theta \\sin\\phi \\\\\n",
    "  z = c + r \\cos\\theta\n",
    "  \\end{cases}\n",
    "  $\n",
    "  （参数范围：$0 \\leq \\theta \\leq \\pi$, $0 \\leq \\phi < 2\\pi$）\n",
    "\n",
    "\n",
    "**②圆柱面**\n",
    "\n",
    "- **直角坐标系方程**：  \n",
    "  $\n",
    "  x^2 + y^2 = r^2\n",
    "  $\n",
    "  （绕 $z$ 轴，半径为 $r$）\n",
    "\n",
    "- **参数方程**：  \n",
    "  $\n",
    "  \\begin{cases}\n",
    "  x = r \\cos\\theta \\\\\n",
    "  y = r \\sin\\theta \\\\\n",
    "  z = z\n",
    "  \\end{cases}\n",
    "  $\n",
    "  （参数范围：$0 \\leq \\theta < 2\\pi$, $z \\in \\mathbb{R}$）\n",
    "\n",
    "\n",
    "**③圆锥面**\n",
    "\n",
    "- **直角坐标系方程**：  \n",
    "  $\n",
    "  x^2 + y^2 = z^2 \\tan^2\\alpha\n",
    "  $\n",
    "  （半顶角为 $\\alpha$，顶点在原点）\n",
    "\n",
    "- **参数方程**：  \n",
    "  $\n",
    "  \\begin{cases}\n",
    "  x = r \\cos\\theta \\\\\n",
    "  y = r \\sin\\theta \\\\\n",
    "  z = r \\cot\\alpha\n",
    "  \\end{cases}\n",
    "  $\n",
    "  （参数范围：$r \\geq 0$, $0 \\leq \\theta < 2\\pi$）\n",
    "\n",
    "\n",
    "**④椭圆抛物面**\n",
    "\n",
    "- **直角坐标系方程**：  \n",
    "  $\n",
    "  z = \\frac{x^2}{a^2} + \\frac{y^2}{b^2}\n",
    "  $\n",
    "\n",
    "**⑤双曲抛物面（马鞍面）**\n",
    "- **直角坐标系方程**：  \n",
    "  $\n",
    "  z = \\frac{x^2}{a^2} - \\frac{y^2}{b^2}\n",
    "  $\n",
    "\n",
    "\n",
    "**⑥单叶双曲面**\n",
    "\n",
    "- **直角坐标系方程**：  \n",
    "  $\n",
    "  \\frac{x^2}{a^2} + \\frac{y^2}{b^2} - \\frac{z^2}{c^2} = 1\n",
    "  $\n",
    "\n",
    "- **参数方程**：  \n",
    "  $\n",
    "  \\begin{cases}\n",
    "  x = a \\cos\\phi\\sec\\theta \\\\\n",
    "  y = b \\sin\\phi\\sec\\theta \\\\\n",
    "  z = c\\tan\\theta\n",
    "  \\end{cases}\n",
    "  $\n",
    "  （参数范围：$0\\lt\\theta\\lt\\frac{\\pi}{2}$, $0 \\lt\\phi< 2\\pi$）\n",
    "\n",
    "\n",
    "**⑦椭球面**\n",
    "\n",
    "- **直角坐标系方程**：  \n",
    "  $\n",
    "  \\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2} = 1\n",
    "  $\n",
    "\n",
    "- **参数方程**：  \n",
    "  $\n",
    "  \\begin{cases}\n",
    "  x = a \\cos\\theta \\cos\\phi \\\\\n",
    "  y = b \\cos\\theta \\sin\\phi \\\\\n",
    "  z = c \\sin\\theta\n",
    "  \\end{cases}\n",
    "  $\n",
    "  （参数范围：$0 \\leq \\theta \\leq \\pi$, $0 \\leq \\phi < 2\\pi$）\n",
    "\n",
    "**⑧双叶双曲面**\n",
    "\n",
    "- **直角坐标系方程**：  \n",
    "  $\n",
    "  \\frac{x^2}{a^2} + \\frac{y^2}{b^2} - \\frac{z^2}{c^2} = -1\n",
    "  $\n",
    "\n",
    "- **参数方程**：  \n",
    "  $\n",
    "  \\begin{cases}\n",
    "  x = a \\cos\\phi\\tan\\theta \\\\\n",
    "  y = b\\sin\\phi\\tan\\theta \\\\\n",
    "  z = csec\\theta\n",
    "  \\end{cases}\n",
    "  $\n",
    "  （参数范围：$0\\lt\\theta\\lt\\pi$, $0 \\phi < 2\\pi$）\n",
    "\n",
    "\n",
    "**补充说明**\n",
    "\n",
    "- **参数方程的意义**：参数 $\\theta$ 和 $\\phi$ 通常对应极角和方位角，$u$ 和 $v$ 可灵活选择以简化方程。\n",
    "- **应用场景**：这些方程广泛用于几何建模、物理问题（如电磁场）和工程设计（如曲面拟合）。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "756e89db-162a-4a41-90f2-09ece8036197",
   "metadata": {},
   "source": [
    "## 概率论"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "97f652ea-bbd5-43d9-b6eb-2a41251a003c",
   "metadata": {},
   "source": [
    "- 对任意两个事件$A,B$,有$P(A-B) = P(A) - P(AB)$;\n",
    "- 对任意两个事件$A,B$,则有$P(A\\cup B) = P(A) + P(B) - P(AB)$;\n",
    "- 对任意三个事件$A,B,C$,有$P(A\\cup B\\cup C) = P(A) + P(B) + P(C) - P(AB) - P(AC) - P(BC) + P(ABC)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "497cad11-bfe1-4e84-b22d-f08481145630",
   "metadata": {},
   "source": [
    "(1)甲乙分别参加不同公司的面试，通过概率分别是$\\frac{1}{7}$和$\\frac{1}{5}$,求至少有一个人通过的概率是：$\\frac{11}{35}$\n",
    "\n",
    "P219"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9836b29f-f564-4cfc-b216-8efab84b579a",
   "metadata": {},
   "source": [
    "(2)一个袋子里有8个黑球，8个白球，随机不放回地连续取球5次，每次取出一个球，求最多取到3个白球的概率。\n",
    "\n",
    "解题思路：最多取到3个球的对立事件是取到4个白球或5个白球。设取到白球的数量为X,$P(X=4) = \\frac{C_8^1C_8^4}{C_{16}^5},P(X=5) = \\frac{C_8^5}{C_{16}^5}$,取到最多取到3个白球的概率为$P(x\\le 3)  = 1- P(X=4) - P(X=5)$。\n",
    "\n",
    "P220"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c869fbc3-77ed-4d6d-b037-bfaf9a9e7d1b",
   "metadata": {},
   "source": [
    "判断某个函数能否成为分布函数的充要条件：①单调性；②有界性；③右连续型。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fff254ce-91f5-4f77-afa2-09e0c9e5e9ee",
   "metadata": {},
   "source": [
    "**常用分布的数学期望和方差**\n",
    "\n",
    "| 分布 | 分布列$p_k$或分布函数$p(x)$ | 期望|方差|\n",
    "|------|--------|---:|---:|\n",
    "|两点分布$B(1,p)$|$p_k = p^k(1-p)^{1-k}, k=0,1$|$p$|$p(1-p)$|\n",
    "|两项分布$B(n,p)$|$p_k = p^k(1-p)^{n-k}, k=0,1,\\cdots,n$|$np$|$np(1-p)$|\n",
    "|泊松分布$P(\\lambda)$|$p_k = \\frac{\\lambda^k}{k!}e^{-\\lambda}, k=0,1,\\cdots,n(\\lambda\\gt 0)$|$\\lambda$|$\\lambda$|\n",
    "|几何分布$G(p)$|$p_k = (1-p)^{k-1}p, k=0,1,\\cdots$|$\\frac{1}{p}$|$\\frac{1-p}{p^2}$|\n",
    "|超几何分布$H(n,N,M)$|$p_k = \\frac{C_M^kC_{N-M}^{n-k}}{C_N^n}, k=0,1,\\cdots,\\min(M,n)$|$n\\frac{M}{N}$|$\\frac{nM(N-M)(N-n)}{N^2(N-1)}$|\n",
    "|均匀分布$U(a,b)$|$f(x) = \\frac{1}{b-a}, a\\lt x\\lt b$|$\\frac{a+b}{2}$|$\\frac{(b-a)^2}{12}$|\n",
    "|指数分布$E(\\lambda)$|$f(x) = \\lambda e^{-\\lambda x}, x\\gt 0$|$\\lambda$|$\\lambda$|\n",
    "|正态分布$N(\\mu,\\sigma^2)$|$f(x) = \\frac{1}{\\sqrt{2\\pi}\\sigma}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}, -\\infty\\lt x\\lt+\\infty$|$\\mu$|$\\sigma^2$|"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dcf71fe7-ef48-433a-8aa2-c62d1eca0a05",
   "metadata": {},
   "source": [
    "## 教学设计"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "39ca4c4b-e61c-4b3f-aa23-31e5f2c06a6a",
   "metadata": {},
   "source": [
    "**教学目标**\n",
    "\n",
    "- <mark>具体学习的知识点</mark>\n",
    "- 经历整个探索过程，感受知识与知识间的紧密联系，增强应用意识，树立学好数学的信心，体会数学的价值；\n",
    "- 通过对公式的推导过程的探索，提高分析问题和解决问题的能力，发展类比推理的意识；\n",
    "- 通过自主探索、合作交流，提升数学兴趣，体会数学的严谨美；"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f130a3f6-31bd-4439-90b5-12ba3c5ae1ae",
   "metadata": {},
   "source": [
    "**教学过程**\n",
    "\n",
    "- ①复习旧知，引入新知(新课导入)\n",
    "    - 引入之前学习的大致内容，通过提问等方式帮助学生回忆，并且顺势衔接到新知识点。\n",
    "    - 【设计意图】\n",
    "        - 教师带领学生复习相关旧知，降低新知的认识难度，为新知的学习做好准备。\n",
    "        - 通过复习旧知，淡化学生对新识的陌生感，降低学生对新知的认知维度，快读进入新课学习中。\n",
    "\n",
    "     \n",
    "- ②提出问题，探索新知（新课讲授）\n",
    "    - 讲授新课内容，提出问题，让学生独立思考、合作交流、探索创新。\n",
    "    - 【设计意图】\n",
    "        - 从一个具体的数学问题入手，引导学生利用已有的知识探索并解决问题，建立前后知识的联系，提升学生的几何直观和数学运算的能力，同时也发展了学生的应用意识。\n",
    "        - 通过小组讨论的方式，学生经历公式的推导过程，从而对教学重点内容印象更加深刻，提升自主探索和合作交流的能力，发展学生的推理意识。\n",
    "\n",
    "      \n",
    "- ③巩固新知，提升练习（巩固练习）\n",
    "    - 布置一些课堂练习，强化巩固，提问答疑。\n",
    "    - 【设计意图】\n",
    "      - 利用习题不仅可以提升对某个知识点（这里应当写出题目中的具体知识点）理解，还以让学习发现某些知识点（这里应当写出题目中的具体知识点）之间的联系。\n",
    "      - 强化学生对某个知识点的理解，帮助学习巩固某个知识点。\n",
    "        \n",
    "         \n",
    "- ④归纳小结，布置作业（小结回顾）\n",
    "    - 教师带领老师对本节课程内容进行反思回顾，构建知识网络，领悟思想方法，布置适量课后习题。\n",
    "    - 【设计意图】：对新知识进一步整合总结，可以帮助学生进一步理解和掌握新知。通过课后练习，让学生进一步地思考和运用新知。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fc92f810-60bc-4ce8-800c-13080db7422b",
   "metadata": {},
   "source": [
    "## 错题"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a4985dd1-53e0-4d4c-9cca-d72dec179c07",
   "metadata": {},
   "source": [
    "1.已知向量组$\\alpha_1 = (3,2,1)^T,\\alpha_2=(3,1,4)^T,\\alpha_3=(1,1,0)^T,\\alpha_4=(8,8,6)^T$。\n",
    "\n",
    "(2) 求向量$\\alpha_4$在基地$\\alpha_1,\\alpha_2,\\alpha_3$下的坐标。可得坐标为(-6,3,17)。\n",
    "\n",
    "来源：2023年上半年简答题10"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9001732d-bc00-4fc8-b9a6-477bb4a5392f",
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   "source": [
    "以下是通过几何方法推导椭圆方程的过程：\n",
    "\n",
    "**定义与设定**\n",
    "\n",
    "- 椭圆的定义为平面内到两个定点$F_1,F_2$的距离之和等于常数（大于$|F_1F_2|$）的点的轨迹。设这两个定点$F_1,F_2$之间的距离为$2c$，动点$M(x,y)$到两定点$F_1,F_2$的距离之和为$2a$（$a>c>0$）。\n",
    "- 以线段$F_1F_2$的中点为原点，$F_1,F_2$所在直线为$x$轴，建立平面直角坐标系。则$F_1(-c,0)$，$F_2(c,0)$。\n",
    "\n",
    "**推导过程**\n",
    "- 根据椭圆的定义，$\\vert MF_1\\vert+\\vert MF_2\\vert = 2a$。\n",
    "- 由两点间距离公式可得：$\\vert MF_1\\vert=\\sqrt{(x + c)^2+y^2}$，$\\vert MF_2\\vert=\\sqrt{(x - c)^2+y^2}$。\n",
    "- 所以$\\sqrt{(x + c)^2+y^2}+\\sqrt{(x - c)^2+y^2}=2a$。\n",
    "- 移项可得$\\sqrt{(x + c)^2+y^2}=2a-\\sqrt{(x - c)^2+y^2}$。\n",
    "- 两边平方可得$(x + c)^2+y^2=4a^2-4a\\sqrt{(x - c)^2+y^2}+(x - c)^2+y^2$。\n",
    "- 展开并化简得：$x^2 + 2cx + c^2+y^2=4a^2-4a\\sqrt{(x - c)^2+y^2}+x^2-2cx + c^2+y^2$，即$4cx - 4a^2=-4a\\sqrt{(x - c)^2+y^2}$。\n",
    "- 两边同时除以$4$得$cx - a^2=-a\\sqrt{(x - c)^2+y^2}$。\n",
    "- 再次两边平方可得$(cx - a^2)^2=a^2[(x - c)^2+y^2]$。\n",
    "- 展开得$c^2x^2-2a^2cx + a^4=a^2(x^2-2cx + c^2+y^2)$。\n",
    "- 继续展开并化简得$c^2x^2-2a^2cx + a^4=a^2x^2-2a^2cx + a^2c^2+a^2y^2$。\n",
    "- 移项合并同类项得$(c^2 - a^2)x^2+a^2y^2=a^2(c^2 - a^2)$。\n",
    "- 令$b^2=a^2 - c^2$（$b>0$），则方程可化为$\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1$（$a>b>0$），这就是椭圆的标准方程。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "268281aa-74a8-4674-a165-ae74a8fece33",
   "metadata": {},
   "source": [
    "**分类是一种重要的数学方法，请简述分类的原则和学习分类的意义。**\n",
    "\n",
    "- 一般地，分类应保证“不重不漏”，因此在分类是应该遵循一下原则：\n",
    "    - 同一性原则。分类应该按照统一标准进行，即每次分类不能同时适用几个不同的分类依据。\n",
    "    - 互斥性原则。分类后的每个情况应当不相容，也就是分类后不能有一些事物既属于这个情况，有属于另一个情况。\n",
    "    - 层次性原则。分类有一次分类和多次分类之分，一次分类是对被讨论对象只分类一次；多次分类是把分类后所得的情况再进行分类，直至满足需要为止，分层不能越级。\n",
    "- 学习分类的意义：\n",
    "    - 分类需要对客观事物进行分析、比较、发现并抽象归纳出事物的一般特性或本质属性，这就为发展数学抽象的核心素养铺平了道路。\n",
    "    - 不同类别的分类为深入认识指明了可能的途径，从而可以由特殊到一般逐渐深入地研究对象。\n",
    "    - 由于分类活动都是从辨别开始的，再抽象为具体概念和定义性概念，最后形成规则和高级规则，即为达到高级思维奠定基础。\n",
    "    - 运用分类思想能够帮助学生有条理、有顺序、不重复、不遗漏地归纳整理知识，形成完善合理的知识网络图。\n",
    "    - 学会分类是组织策略的重要前提，这是由于组织策略是根据知识经验之间的内在关系，对学习材料进行系统、有序的分类，整理和概括，使之结构合理化。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4164c77-e26e-44b9-ae2e-979488397467",
   "metadata": {},
   "source": [
    "**四基和四能**\n",
    "\n",
    "- 四基：数学基础知识、基本技能、基本思想、基本活动经验的简称；\n",
    "- 四能：从数学角度发现和提出问题的能力、分析和解决问题的能力的简称。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "af940cf7-efeb-4762-ab7a-1c86cd52c6c8",
   "metadata": {},
   "source": []
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